Chapter 12 Resource Masters - Math Problem Solving - …©Glencoe/McGraw-Hill v Glencoe Geometry Assessment Options The assessment masters in the Chapter 12 Resources Mastersoffer

Chapter 12Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 12 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860189-4 GeometryChapter 12 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 12-1Study Guide and Intervention . . . . . . . . 661–662Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 663Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 664Reading to Learn Mathematics . . . . . . . . . . 665Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 666







Chapter 12 AssessmentChapter 12 Test, Form 1 . . . . . . . . . . . 703–704Chapter 12 Test, Form 2A . . . . . . . . . . 705–706Chapter 12 Test, Form 2B . . . . . . . . . . 707–708Chapter 12 Test, Form 2C . . . . . . . . . . 709–710Chapter 12 Test, Form 2D . . . . . . . . . . 711–712Chapter 12 Test, Form 3 . . . . . . . . . . . 713–714Chapter 12 Open-Ended Assessment . . . . . 715Chapter 12 Vocabulary Test/Review . . . . . . 716Chapter 12 Quizzes 1 & 2 . . . . . . . . . . . . . . 717Chapter 12 Quizzes 3 & 4 . . . . . . . . . . . . . . 718Chapter 12 Mid-Chapter Test . . . . . . . . . . . . 719Chapter 12 Cumulative Review . . . . . . . . . . 720Chapter 12 Standardized Test Practice 721–722

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 12 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 12 Resource Masters includes the core materialsneeded for Chapter 12. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 12-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Geometry

Assessment OptionsThe assessment masters in the Chapter 12Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 684–685. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

1212

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 12. As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Geometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

altitude

axis

cone

cylinder

great circle

hemisphere

lateral area

lateral edges

lateral faces

net

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

oblique

orthogonal drawing

ohr·THAHG·uhn·uhl

polyhedrons

prism

PRIZ·uhm

pyramid

regular polyhedron

regular prism

right

sphere

SFIR

surface area

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

1212

Study Guide and InterventionThree-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

© Glencoe/McGraw-Hill 661 Glencoe Geometry

Less

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Drawings of Three-Dimensional Figures To work with a three-dimensionalobject, a useful skill is the ability to make an orthogonal drawing, which is a set of two-dimensional drawings of the different sides of the object. For a square pyramid, you wouldshow the top view, the left view, the front view, and the right view.

Draw the back view of the figure given the orthogonal drawing.• The top view indicates two columns.• The left view indicates that the height of figure is

three blocks.• The front view indicates that the columns have heights 2 and 3 blocks.• The right view indicates that the height of the figure is three blocks.

Use blocks to make a model of the object. Then use your model to draw the back view. The back view indicates that the columns have heights 3 and 2 blocks.

Draw the back view and corner view of a figure given each orthogonal drawing.

1. 2.

3. 4.

back viewcorner view

top view left view front view right view







back viewobject


object top view left view front view right view

ExercisesExercises

ExampleExample


Identify Three-Dimensional Figures A polyhedron is a solid with all flatsurfaces. Each surface of a polyhedron is called a face, and each line segment where facesintersect is called an edge. Two special kinds of polyhedra are prisms, for which two facesare congruent, parallel bases, and pyramids, for which one face is a base and all the otherfaces meet at a point called the vertex. Prisms and pyramids are named for the shape oftheir bases, and a regular polyhedron has a regular polygon as its base.

Other solids are a cylinder, which has congruent circular bases in parallel planes, a cone,which has one circular base and a vertex, and a sphere.

Identify each solid. Name the bases, faces, edges, and vertices.

pentagonalprism

squarepyramid

pentagonalpyramid

rectangularprism

sphereconecylinder

Study Guide and Intervention (continued)

Three-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

ExampleExamplea.

The figure is a rectangular pyramid. The base isrectangle ABCD, and the four faces �ABE, �BCE,�CDE, and �ADE meet at vertex E. The edges areA�B�, B�C�, C�D�, A�D�, A�E�, B�E�, C�E�, and D�E�. The verticesare A, B, C, D, and E.

E

A B

CD

b.

This solid is a cylinder. Thetwo bases are �O and �P.

P

O

ExercisesExercises


1. 2.

3. 4.

A B

CF

D E

P WQ

VZ X

U

SR

Y

T

S

R

Skills PracticeThree-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1


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1


1. 2.


3.

4.

5.

S

R

F

AB C

DE

U T

WV

R S

XY

back view corner view






1. 2.


3.

4.

5. MINERALS Pyrite, also known as fool’s gold, can form crystals that are perfect cubes.Suppose a gemologist wants to cut a cube of pyrite to get a square and a rectangularface. What cuts should be made to get each of the shapes? Illustrate your answers.

Z

T

I J

MH K

N

O L





Practice Three-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

Reading to Learn MathematicsThree-Dimensional Figures

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1


Less

on

12-

1

Pre-Activity Why are drawings of three-dimensional structures valuable toarcheologists?

Read the introduction to Lesson 12-1 at the top of page 636 in your textbook.

Why do you think archeologists would want to know the details of the sizesand shapes of ancient three-dimensional structures?

Reading the Lesson1. Match each description from the first column with one of the terms from the second

column. (Some of the terms may be used more than once or not at all.)

a. a polyhedron with two parallel congruent basesb. the set of points in space that are a given distance from a

given pointc. a regular polyhedron with eight facesd. a polyhedron that has all faces but one intersecting at one

pointe. a line segment where two faces of a polyhedron intersectf. a solid with congruent circular bases in a pair of parallel

planesg. a regular polyhedron whose faces are squaresh. a flat surface of a polyhedron

i. octahedronii. face

iii. icosahedroniv. edgev. prism

vi. dodecahedronvii. cylinderviii. sphereix. conex. hexahedron

xi. pyramidxii. tetrahedron

2. Fill in the missing numbers, words, or phrases to complete each sentence.

a. A triangular prism has vertices. It has faces: bases are congruent

, and faces are parallelograms.

b. A regular octahedron has vertices and faces. Each face is a(n)

.

c. A hexagonal prism has vertices. It has faces: of them are the bases,

which are congruent , and the other faces are parallelograms.

d. An octagonal pyramid has vertices and faces. The base is a(n)

, and the other faces are .

e. There are exactly types of regular polyhedra. These are called the

solids. A polyhedron whose faces are regular pentagons is a

, which has faces.

Helping You Remember3. A good way to remember the characteristics of geometric solids is to think about how

different solids are alike. Name a way in which pyramids and cones are alike.


Drawing Solids on Isometric Dot PaperIsometric dot paper is helpful for drawing solids. Remember to usedashed lines for hidden edges.

For each solid shown, draw another solid whose dimensions are twice as large.

1. 2.

3. 4.

5. 6.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

Study Guide and InterventionNets and Surface Area

NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2


Less

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Models for Three-Dimensional Figures One way to relate a three-dimensionalfigure and a two-dimensional drawing is to use isometric dot paper. Another way is to makea flat pattern, called a net, for the surfaces of a solid.

Use isometric dot paper to sketch a triangular prism with 3-4-5 right triangles as bases and with a height of 3 units.Step 1 Draw A�B� at 3 units and draw A�C� at 4 units.Step 2 Draw A�D�, B�E�, and C�F�, each at 3 units.Step 3 Draw B�C� and �DEF.

Match the net at the right with one of the solids below.

The six squares of the net can be folded into a cube. The net represents solid c.

Sketch each solid using isometric dot paper.

1. cube with edge 4 2. rectangular prism 1 unit high, 5 unitslong, and 4 units wide

Draw a net for each solid.

3. 4.

5. 6.

c.b.a.

A

B CD

E F

ExercisesExercises

Example 1Example 1

Example 2Example 2


Surface Area The surface area of a solid is the sum of the areas of the faces of thesolid. Nets are useful in visualizing each face and calculating the area of the faces.

Find the surface area of the triangular prism.

First draw a net using rectangular dot paper. Using the Pythagorean Theorem,

the hypotenuse of the right triangle is �32 � 4�2� or 5.

Surface area � A � B � C � D � E

� �12�(4 � 3) � 4 � 4 � 4 � 3 � 4 � 5 � �

12�(4 � 3)

� 60 square units

Find the surface area of each solid. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

4025

2020

1518

88

10

9126

5 m5 m

5 m

24 cm5 cm

10 cm

12 in.8 in.

5 in.

A

C

E

B D4

43


Nets and Surface Area

NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2

ExercisesExercises

ExampleExample

Skills PracticeNets and Surface Area

NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2


Less

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2


1. cube 2 units on each edge 2. rectangular prism 2 units high,5 units long, and 2 units wide

For each solid, draw a net and find the surface area.

3.

4.

55

6

7

3

5



1. rectangular prism 3 units high, 2. triangular prism 3 units high, whose bases 3 units long, and 2 units wide are right triangles with legs 2 units and

4 units long

3. For the solid, draw a net and find the surface area.

4. SHIPPING Rawanda needs to wrap a package to ship to her aunt. The rectangularpackage measures 2 inches high, 10 inches long, and 4 inches wide. Draw a net of thepackage. How much wrapping paper does Rawanda need to cover the package?

2

10

2

4

3

96

15

Practice Nets and Surface Area

NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2

Reading to Learn MathematicsNets and Surface Area

NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2


Less

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2

Pre-Activity Why is surface area important to car manufacturers?


Suppose you are a passenger in a car that is moving at 50 miles per hour onthe highway. If you hold your right hand just outside the right window, inwhat position does your hand encounter the greatest air resistance?

Reading the Lesson

1. Name the solid that can be formed by folding each net without any overlap.

a. b.

c. d.

2. Supply the missing number, word, or phrase to make a true statement. Be as specific aspossible.

a. To find the surface area of a cube, add the areas of congruent squares.

b. To find the surface area of a cylinder, add the areas of two congruent

and one .

c. To find the surface area of a dodecahedron, add the areas of 12 congruent

.

d. To find the surface area of an icosahedron, add the areas of congruent

triangles.

Helping You Remember

3. A good way to remember a mathematical concept is to think about how the conceptrelates to everyday life. How could you explain the concept of surface area using aneveryday situation?


Pyramids

1. On a sheet of paper, draw the figure at the right so that the measurements are accurate.Cut out the shape and fold it to make a pyramid.

2. Measure the height from the highest point straight down (perpendicular) to the base.What is the height of the pyramid?

The drawing at the right shows the pyramid you made.Answer each of the following.

3. Measure the length of y in your pyramid.

4. Measure the length of z, the height of one of the triangular faces.

5. Use the Pythagorean Theorem to find x, the height of the pyramid.Is this equal to the height of your model?

Make a paper pattern for each pyramid below. Cut out your pattern and fold it tomake a model. Measure the height of the pyramid and use the PythagoreanTheorem to verify this measure.

6. 7.

12.5 cm

7 cm

5 cm

8 cm

y

x z

10 cm

13cm

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2

Study Guide and InterventionSurface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3


Less

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Lateral Areas of Prisms Here are some characteristics of prisms.

• The bases are parallel and congruent.• The lateral faces are the faces that are not bases.• The lateral faces intersect at lateral edges, which are parallel.• The altitude of a prism is a segment that is perpendicular

to the bases with an endpoint in each base.• For a right prism, the lateral edges are perpendicular to the

bases. Otherwise, the prism is oblique.

Lateral Area If a prism has a lateral area of L square units, a height of h units, of a Prism and each base has a perimeter of P units, then L � Ph.

Find the lateral area of the regular pentagonal prism above if eachbase has a perimeter of 75 centimeters and the altitude is 10 centimeters.L � Ph Lateral area of a prism

� 75(10) P � 75, h � 10

� 750 Multiply.

The lateral area is 750 square centimeters.

Find the lateral area of each prism.

1. 2.

3. 4.

5. 6.

4 m16 m

4 in.

4 in.

12 in.

20 cm

10 cm 10 cm

12 cm9 cm

6 in.18 in.

15 in.

10 in.

8 in.4 m

3 m10 m

pentagonal prism

altitude

lateraledge lateral

face

ExampleExample

ExercisesExercises


Surface Areas of Prisms The surface area of a prism is thelateral area of the prism plus the areas of the bases.

Surface AreaIf the total surface area of a prism is T square units, its height is

of a Prismh units, and each base has an area of B square units and a perimeter of P units, then T � L � 2B.

Find the surface area of the triangular prism above.Find the lateral area of the prism.

L � Ph Lateral area of a prism

� (18)(10) P � 18, h � 10

� 180 cm2 Multiply.

Find the area of each base. Use the Pythagorean Theorem to find the height of thetriangular base.

h2 � 32 � 62 Pythagorean Theorem

h2 � 27 Simplify.

h � 3�3� Take the square root of each side.

B � �12� � base � height Area of a triangle

� �12�(6)(3�3�) or 15.6 cm2

The total area is the lateral area plus the area of the two bases.

T � 180 � 2(15.6) Substitution

� 211.2 cm2 Simplify.

Find the surface area of each prism. Round to the nearest tenth if necessary.

1. 2. 3.

4. 5. 6.

8 m

8 m8 m

8 in.

8 in.

8 in.

15 in.6 in.

12 in.

12 m

5 m6 m

3 m6 m

4 m

24 in.

10 in.

8 in.

6 cm

6 cm

6 cm

10 cmh


Surface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3


Less

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Find the lateral area of each prism.

1. 2.

3. 4.


5. 6.

7. 8.

7

4

39

610

15

7

13

18

8

6

9

9

9

12

105

6

8

8

6

12

12

12

10


Find the lateral area of each prism. Round to the nearest tenth if necessary.

1. 2.

3. 4.


5. 6.

7. 8.

9. CRAFTS Becca made a rectangular jewelry box in her art class and plans to cover it in red

silk. If the jewelry box is 6�12� inches long, 4 �

12� inches wide, and 3 inches high, find the

surface area that will be covered.

14

14

23

9

95

6

13

79

4

17

4

4

9.5

52

11

8

105

15

15

32

Practice Surface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

Reading to Learn MathematicsSurface Areas of Prisms

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3


Less

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Pre-Activity How do brick masons know how many bricks to order for a project?


How could the brick mason figure out approximately how many bricks areneeded for the garage?

Reading the Lesson1. Determine whether each sentence is always, sometimes, or never true.

a. A base of a prism is a face of the prism.b. A face of a prism is a base of the prism.c. The lateral faces of a prism are rectangles.d. If a base of a prism has n vertices, then the prism has n faces.e. If a base of a prism has n vertices, then the prism has n lateral edges.f. In a right prism, the lateral edges are also altitudes.g. The bases of a prism are congruent regular polygons.h. Any two lateral edges of a prism are perpendicular to each other.i. In a rectangular prism, any pair of opposite faces can be called the bases.j. All of the lateral faces of a prism are congruent to each other.

2. Explain the difference between the lateral area of a prism and the surface area of aprism. Your explanation should apply to both right and oblique prisms. Do not use anyformulas in your explanation.

3. Refer to the figure.

a. Name this solid with as specific a name as possible.

b. Name the bases of the solid.c. Name the lateral faces.

d. Name the edges.e. Name an altitude of the solid.f. If a represents the area, P represents the perimeter of one of the bases, and x � AF,

write an expression for the surface area of the solid that involves a, P, and x.

Helping You Remember4. A good way to remember a new mathematical term is to relate it to an everyday use of

the same word. How can the way the word lateral is used in sports help you rememberthe meaning of the lateral area of a solid?

A

B C

D

IF

E

G H

J


Cross Sections of PrismsWhen a plane intersects a solid figure to form a two-dimensional figure, the results is called a cross section. The figure at the right shows a plane intersecting a cube. The cross section is a hexagon.

For each right prism, connect the labeled points in alphabetical order to show a cross section. Then identify the polygon.

1. 2. 3.

Refer to the right prisms shown at the right. In the rectangular prism, A and Care midpoints. Identify the cross-section polygon formed by a plane containing the given points.

4. A, C, H

5. C, E, G

6. H, C, E, F

7. H, A, E

8. B, D, F

9. V, X, R

10. R, T, Y

11. R, S, W

A

B

H G

F

DC

E

V X

SU

Y

Z

R

T

W

A B

CD

X

Y

Z

S

R

U

T

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

Study Guide and InterventionSurface Areas of Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4


Less

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Lateral Areas of Cylinders A cylinder is a solid whose bases are congruent circles that lie in parallel planes. The axisof a cylinder is the segment whose endpoints are the centers of these circles. For a right cylinder, the axis and the altitude of the cylinder are equal. The lateral area of a right cylinder is thecircumference of the cylinder multiplied by the height.

Lateral Area If a cylinder has a lateral area of L square units, a height of h units, of a Cylinder and the bases have radii of r units, then L � 2�rh.

Find the lateral area of the cylinder above if the radius of the baseis 6 centimeters and the height is 14 centimeters.L � 2�rh Lateral area of a cylinder

� 2�(6)(14) Substitution

� 527.8 Simplify.

The lateral area is about 527.8 square centimeters.

Find the lateral area of each cylinder. Round to the nearest tenth.

1. 2.

3. 4.

5. 6. 2 m

1 m12 m

4 m

20 cm

8 cm

6 cm

3 cm

3 cm

6 in.10 in.

12 cm

4 cm

radius of baseaxis

base

baseheight

ExercisesExercises

ExampleExample


Surface Areas of Cylinders The surface area of a cylinder is the lateral area of the cylinder plus the areas of the bases.

Surface Area If a cylinder has a surface area of T square units, a height of of a Cylinder h units, and the bases have radii of r units, then T � 2�rh � 2�r 2.

Find the surface area of the cylinder.Find the lateral area of the cylinder. If the diameter is 12 centimeters, then the radius is 6 centimeters.

L � Ph Lateral area of a cylinder

� (2�r)h P � 2�r

� 2�(6)(14) r � 6, h � 14

� 527.8 Simplify.

Find the area of each base.

B � �r2 Area of a circle

� �(6)2 r � 6

� 113.1 Simplify.

The total area is the lateral area plus the area of the two bases.T � 527.8 � 113.1 � 113.1 or 754 square centimeters.

Find the surface area of each cylinder. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

20 in.

8 in.

2 m 15 m

8 in.

12 in.

2 yd

3 yd

2 m

2 m

12 in.

10 in.

14 cm

12 cm

base

baselateral area


Surface Areas of Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4

ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4


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Find the surface area of a cylinder with the given dimensions. Round to thenearest tenth.

1. r � 10 in., h � 12 in. 2. r � 8 cm, h � 15 cm

3. r � 5 ft, h � 20 ft 4. d � 20 yd, h � 5 yd

5. d � 8 m, h � 7 m 6. d � 24 mm, h � 20 mm


7. 8.

Find the radius of the base of each cylinder.

9. The surface area is 603.2 square meters, and the height is 10 meters.

10. The surface area is 100.5 square inches, and the height is 6 inches.

11. The surface area is 226.2 square centimeters, and the height is 5 centimeters.

12. The surface area is 1520.5 square yards, and the height is 14.2 yards.

4 m

8.5 m

5 ft

7 ft


Find the surface area of a cylinder with the given dimensions. Round to thenearest tenth.

1. r � 8 cm, h � 9 cm 2. r � 12 in., h � 14 in.

3. d � 14 mm, h � 32 mm 4. d � 6 yd, h � 12 yd

5. r � 2.5 ft, h � 7 ft 6. d � 13 m, h � 20 m


7. 8.

Find the radius of the base of each right cylinder.

9. The surface area is 628.3 square millimeters, and the height is 15 millimeters.

10. The surface area is 892.2 square feet, and the height is 4.2 feet.

11. The surface area is 158.3 square inches, and the height is 5.4 inches.

12. KALEIDOSCOPES Nathan built a kaleidoscope with a 20-centimeter barrel and a 5-centimeter diameter. He plans to cover the barrel with embossed paper of his owndesign. How many square centimeters of paper will it take to cover the barrel of thekaleidoscope?

12 m30 m

19 in.

17 in.

Practice Surface Areas of Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsSurface Areas of Cylinders

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Pre-Activity How are cylinders used in extreme sports?Read the introduction to Lesson 12-4 at the top of page 655 in your textbook.

Why is the surface area of a half-pipe more than half the surface area of acomplete pipe?

Reading the Lesson1. Underline the correct word or phrase to form a true statement.

a. The bases of a cylinder are (rectangles/regular polygons/circles).

b. The (axis/radius/diameter) of a cylinder is the segment whose endpoints are thecenters of the bases.

c. The net of a cylinder is composed of two congruent (rectangles/circles) and one(rectangle/semicircle).

d. In a right cylinder, the axis of the cylinder is also a(n) (base/lateral edge/altitude).

e. A cylinder that is not a right cylinder is called an (acute/obtuse/oblique) cylinder.

2. Match each description from the first column with an expression from the second columnthat represents its value.a. the lateral area of a right cylinder in which the radius of

each base is x cm and the length of the axis is y cm

b. the surface area of a right prism with square bases in whichthe length of a side of a base is x cm and the length of alateral edge is y cm

c. the surface area of a right cylinder in which the radius of abase is x cm and the height is y cm

d. the surface area of regular hexahedron (cube) in which thelength of each edge is x cm

e. the lateral area of a triangular prism in which the bases areequilateral triangles with side length x cm and the height isy cm

f. the surface area of a right cylinder in which the diameter ofthe base is x cm and the length of the axis is y cm

i. (2x2 � 4xy) cm2

ii. (2�xy � 2�x2) cm2

iii. 3xy cm2

iv. 6x2 cm2

v. 2�xy cm2

vi. ��2x2� � �xy� cm2

Helping You Remember3. Often the best way to remember a mathematical formula is to think about where the

different parts of the formula come from. How can you use this approach to remember theformula for the surface area of a cylinder?


Can-didly SpeakingBeverage companies have mathematically determined the ideal shape for a 12-ounce soft-drink can. Why won't any shape do? Some shapes are moreeconomical than others. In order to hold 12 ounces of soft drink, an extremelyskinny can would have to be very tall. The total amount of aluminum used forsuch a shape would be greater than the amount used for the conventional shape,thus costing the company more to make the skinny can. The radius r chosendetermines the height h needed to hold 12 ounces of liquid. This also determinesthe amount of aluminum needed to make the can. Companies also have to keepin mind that the top of the can is three times thicker than the bottom and sides.Why? So you won’t tear off the entire top when you open the can! The followingformulas can be used to find the height and amount of aluminum a needed for a12-ounce soft-drink can.

h � �1�

7.r829

�

a � 0.02��r2 �

r35.78�� The values of r and h are measured in inches.

Find the height needed for a 12-ounce can for each radius.Round to the nearest tenth.

1. 2�23� in. 2. 1 in. 3. 3 in. 4. 1�

34� in.

Sketch the shape of each can in Exercises 1–4.

5. Exercise 1 6. Exercise 2 7. Exercise 3 8. Exercise 4

9. a. Measure the radius of a soft-drink can.

b. Use the formula to find the height of the can.

c. Measure the height of a soft-drink can.

d. How does this measure compare to your findings in part a?

10. Find the amount of aluminum used in making a soft-drink can.

13–4 in.

1.9 in.

3 in.

0.6 in.5.7 in.

1 in.22–3 in.

Enrichment

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Study Guide and InterventionSurface Areas of Pyramids

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Lateral Areas of Regular Pyramids Here are some properties of pyramids.• The base is a polygon.• All of the faces, except the base, intersect in a common point known as the vertex.• The faces that intersect at the vertex, which are called lateral faces, are triangles.

For a regular pyramid, the base is a regular polygon and the slant height is the height of each lateral face.

Lateral Area of a If a regular pyramid has a lateral area of L square units, a slant height of � units, Regular Pyramid and its base has a perimeter of P units, then L � �1

2�P�.

The roof of a barn is a regular octagonal pyramid. The base of the pyramid has sides of 12 feet,and the slant height of the roof is 15 feet. Find the lateral area of the roof.The perimeter of the base is 8(12) or 96 feet.

L � �12�P� Lateral area of a pyramid

� �12�(96)(15) P � 96, � � 15

� 720 Multiply.

The lateral area is 720 square feet.

Find the lateral area of each regular pyramid. Round to the nearest tenth ifnecessary.

1. 2.

3. 4.

5. 6.

45� 12 yd60�18 in.

60� 6 ft

42 m

20 m

3.5 ft

10 ft

8 cm

8 cm

8 cm

15 cm

base

lateral edgelateral face

vertex

slant height

ExercisesExercises

ExampleExample


Surface Areas of Regular Pyramids The surface area of a regular pyramid is the lateral area plus the area of the base.

Surface Area of aIf a regular pyramid has a surface area of T square units,

Regular Pyramida slant height of � units, and its base has a perimeter of P units and an area of B square units, then T � �

12

�P� � B.

For the regular square pyramid above, find the surface area to thenearest tenth if each side of the base is 12 centimeters and the height of thepyramid is 8 centimeters.Look at the pyramid above. The slant height is the hypotenuse of a right triangle. One leg ofthat triangle is the height of the pyramid, and the other leg is half the length of a side of thebase. Use the Pythagorean Theorem to find the slant height �.

�2 � 62 � 82 Pythagorean Theorem

� 100 Simplify.

� � 10 Take the square root of each side.

T � �12�P� � B Surface area of a pyramid

� �12�(4)(12)(10) � 122 P � (4)(12), � � 10, B � 122

� 384 Simplify.

The surface area is 384 square centimeters.

Find the surface area of each regular pyramid. Round to the nearest tenth ifnecessary.

1. 2.

3. 4.

5. 6.12 yd

10 yd

12 cm 13 cm

6 in.8.7 in. 15 in.

60�

10 cm

45�

8 ft

15 cm

20 cm

lateral edge

baseslant height height


Surface Areas of Pyramids

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ExampleExample

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Skills PracticeSurface Area of Pyramids

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1. 2.

3. 4.

5. 6.

7. 8.

16 in.

20 in.

18 m

12 m

6 yd

7 yd

9 mm

6 mm

14 ft

12 ft9 m

10 m

20 in.

8 in.4 cm

7 cm



1. 2.

3. 4.

5. 6.

7. 8.

9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of thepyramid has sides of 0.5 meters and the slant height of the roof is 1.9 meters, find thearea of the roof.

4.5 m

12 m

5.2 yd

3 yd

15 mm

12 mm12 in.11 in.

8 cm

2.5 cm

13 ft

5 ft

12 m

7 m9 yd

10 yd

Practice Surface Area of Pyramids

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Reading to Learn MathematicsSurface Areas of Pyramids

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Pre-Activity How are pyramids used in architecture?


Why do you think that the architect for the new entrance to the Louvredecided to use a pyramid rather than a rectangular prism?

Reading the Lesson

1. In the figure, ABCDE has congruent sides and congruent angles.

a. Describe this pyramid with as specific a name as possible.

b. Use the figure to name the base of this pyramid.

c. Describe the base of the pyramid.

d. Name the vertex of the pyramid.

e. Name the lateral faces of the pyramid.

f. Describe the lateral faces.

g. Name the lateral edges of the pyramid.

h. Name the altitude of the pyramid.

i. Write an expression for the height of the pyramid.

j. Write an expression for the slant height of the pyramid.

2. In a regular square pyramid, let s represent the side length of the base, h represent theheight, a represent the apothem, and � represent the slant height. Also, let L representthe lateral area and let T represent the surface area. Which of the followingrelationships are correct?

A. s � 2a B. a2 � �2 � h2 C. L � 4�s

D. h � ��2 � a�2� E. ��2s

��2 � h2 � �2 F. T � s2 � 2�s


3. A good way to remember something is to explain it to someone else. Suppose that one ofyour classmates is having trouble remembering the difference between the height andthe slant height of a regular pyramid. How can you explain this concept?

E D

CAS

B

Q

P


Two Truncated SolidsTo create a truncated solid, you could start with an ordinary solid and then cut off thecorners. Another way to make such a shape is to use the patterns on this page.

The Truncated Octahedron

1. Two copies of the pattern at the right can be used to make a truncated octahedron, a solid with 6 square faces and 8 regular hexagonal faces.

Each pattern makes half of the truncatedoctahedron. Attach adjacent faces using glue or tape to make a cup-shaped figure.

The Truncated Tetrahedron

2. The pattern below will make a truncated tetrahedron,a solid with 8 polygonal faces: 4 hexagons and 4 equilateral triangles.

Solve.

3. Find the surface area of the truncated octahedron if each polygon in the pattern has sides of 3 inches.

4. Find the surface area of the truncated tetrahedron if each polygon in the pattern has sides of 3 inches.

Area Formulas for Regular Polygons(s is the length of one side)

triangle A � �s42��3�

hexagon A � �32s2��3�

octagon A � 2s2(�2� � 1)

Tape orglue here.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionSurface Areas of Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

12-612-6


Less

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6Lateral Areas of Cones Cones have the following properties.• A cone has one circular base and one vertex.• The segment whose endpoints are the vertex and

the center of the base is the axis of the cone.• The segment that has one endpoint at the vertex, is

perpendicular to the base, and has its other endpoint on the base is the altitude of the cone.

• For a right cone the axis is also the altitude, and any segment from the circumference of the base to the vertex is the slant height �. If a cone is not a right cone, it is oblique.

Lateral Area If a cone has a lateral area of L square units, a slant height of � units, of a Cone and the radius of the base is r units, then L � �r�.

Find the lateral area of a cone with slant height of 10 centimeters and a base with a radius of 6 centimeters.L � �r� Lateral area of a cone

� �(6)(10) r � 6, � � 10

� 188.5 Simplify.

The lateral area is about 188.5 square centimeters.

Find lateral area of each circular cone. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

8 yd

16 yd12 in.

5 in.

26 mm

20 mm

6��3 m

60�

9 cm

15 cm

3 cm4 cm

6 cm10 cm

axis

base base� slant height

right coneoblique cone

altitudeV V

ExercisesExercises

ExampleExample


Surface Areas of Cones The surface area of a cone is the lateral area of the cone plus the area of the base.

Surface Area If a cone has a surface area of T square units, a slant height of of a Right Cone � units, and the radius of the base is r units, then T � �r� � �r 2.

For the cone above, find the surface area to the nearest tenth if theradius is 6 centimeters and the height is 8 centimeters.The slant height is the hypotenuse of a right triangle with legs of length 6 and 8. Use thePythagorean Theorem.

�2 � 62 � 82 Pythagorean Theorem

�2 � 100 Simplify.

� � 10 Take the square root of each side.

T � �r� � �r2 Surface area of a cone

� �(6)(10) � � � 62 r � 6, � � 10

� 301.6 Simplify.

The surface area is about 301.6 square centimeters.

Find the surface area of each cone. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

60�

8��3 yd26 m

40 m

4 in.

45�12 cm

13 cm

5 ft

30�9 cm

12 cm

�slant height

height

r


Surface Areas of Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Cones

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Less

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6Find the surface area of each cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. Find the surface area of a cone if the height is 12 inches and the slant height is 15 inches.

8. Find the surface area of a cone if the height is 9 centimeters and the slant height is 12 centimeters.

9. Find the surface area of a cone if the height is 10 meters and the slant height is 14 meters.

10. Find the surface area of a cone if the height is 5 feet and the slant height is 7 feet.

4 yd6 yd

7 cm

22 cm

17 mm

9 mm

8 in.

21 in.

10 ft

25 ft14 m

5 m


Find the surface area of each cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. Find the surface area of a cone if the height is 8 feet and the slant height is 10 feet.

8. Find the surface area of a cone if the height is 14 centimeters and the slant height is16.4 centimeters.

9. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches.

10. HATS Cuong bought a conical hat on a recent trip to central Vietnam. The basic frame ofthe hat is 16 hoops of bamboo that gradually diminish in size. The hat is covered in palmleaves. If the hat has a diameter of 50 centimeters and a slant height of 32 centimeters,what is the lateral area of the conical hat?

7 cm21 cm5 m

4 m

3 ft 19 ft

48 yd

25 yd

7 mm18 mm

9 in.13 in.

Practice Surface Areas of Cones

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Reading to Learn MathematicsSurface Areas of Cones

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6Pre-Activity How is the lateral area of a cone used to cover tepees?


If you wanted to build a tepee of a certain size, how would it help you toknow the formula for the lateral area of a cone?

Reading the Lesson

1.

a. Which net will give the cone with the greatest lateral area?

b. Which net will give the tallest cone?

2. Refer to the figure at the right. Suppose you have removed the circular base of the cone and cut from V to A so that you can unroll the lateralsurface onto a flat table.

a. How can you be sure that the flattened-out piece is a sector of a circle?

b. How do you know that the flattened-out piece is not a full circle?

3. Suppose you have a right cone with radius r, diameter d, height h, and slant height �.Which of the following relationships involving these lengths are correct?

A. r � 2d B. r � h � � C. r2 � h2 � �2

D. r2 � �2 � h2 E. r � ��2 � h�2� F. h � ��2 � r�2�


4. One way to remember a new formula is to relate it to a formula you already know.Explain how the formulas for the lateral areas of a pyramid and a cone are similar.

V

P A B

h

r

�

Net A Net B Net C


Cone Patterns

The pattern at the right is made from a circle. It can be folded to make a cone.

1. Measure the radius of the circle to the nearest centimeter.

2. The pattern is what fraction of the complete circle?

3. What is the circumference of the complete circle?

4. How long is the circular arc that is the outside of the pattern?

5. Cut out the pattern and tape it together to form a cone.

6. Measure the diameter of the circular base of the cone.

7. What is the circumference of the base of the cone?

8. What is the slant height of the cone?

9. Use the Pythagorean Theorem to calculate the height of the cone.Use a decimal approximation. Check your calculation by measuring the height with a metric ruler.

10. Find the lateral area.

11. Find the total surface area.

Make a paper pattern for each cone with the given measurements.Then cut the pattern out and make the cone. Find the measurements.

12. 13.

diameter of base � diameter of base �

lateral area � lateral area �

height of cone � height of cone �(to nearest tenth of a centimeter) (to nearest tenth of a centimeter)

20 cm120�6 cm

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Study Guide and InterventionSurface Areas of Spheres

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Properties of Spheres A sphere is the locus of all points that are a given distance from a given point called its center.

Here are some terms associated with a sphere.• A radius is a segment whose endpoints are the

center of the sphere and a point on the sphere.• A chord is a segment whose endpoints are points

on the sphere.• A diameter is a chord that contains the sphere’s

center.• A tangent is a line that intersects the sphere in

exactly one point.• A great circle is the intersection of a sphere and

a plane that contains the center of the sphere.• A hemisphere is one-half of a sphere. Each great

circle of a sphere determines two hemispheres.

Determine the shapes you get when you intersect a plane with a sphere.

The intersection of plane M The intersection of plane N The intersection of plane Pand sphere O is point P. and sphere O is circle Q. and sphere O is circle O.

A plane can intersect a sphere in a point, in a circle, or in a great circle.

Describe each object as a model of a circle, a sphere, a hemisphere, or none of these.

1. a baseball 2. a pancake 3. the Earth

4. a kettle grill cover 5. a basketball rim 6. cola can

Determine whether each statement is true or false.

7. All lines intersecting a sphere are tangent to the sphere.

8. Every plane that intersects a sphere makes a great circle.

9. The eastern hemisphere of Earth is congruent to the western hemisphere.

10. The diameter of a sphere is congruent to the diameter of a great circle.

OPO

QN

O

PM

R�S� is a radius. A�B� is a chord.S�T� is a diameter. VX�� is a tangent.The circle that contains points S, M, T, and N isa great circle; it determines two hemispheres.

chord tangent

great circle

diameter

radius

S

T

W

RN

B

V

X

A

M

sphere

center

ExampleExample

ExercisesExercises


Surface Areas of Spheres You can think of the surface area of a sphere as the total area of all of the nonoverlapping strips it would take to cover thesphere. If r is the radius of the sphere, then the area of a great circle of thesphere is �r2. The total surface area of the sphere is four times the area of agreat circle.

Surface Area of a Sphere

If a sphere has a surface area of T square units and a radius of r units, then T � 4�r 2.

Find the surface area of a sphere to the nearest tenth if the radius of the sphere is 6 centimeters.T � 4�r2 Surface area of a sphere

� 4� � 62 r � 6

� 452.4 Simplify.

The surface area is 452.4 square centimeters.

Find the surface area of each sphere with the given radius or diameter to thenearest tenth.

1. r � 8 cm 2. r � 2�2� ft

3. r � � cm 4. d � 10 in.

5. d � 6� m 6. d � 16 yd

7. Find the surface area of a hemisphere with radius 12 centimeters.

8. Find the surface area of a hemisphere with diameter � centimeters.

9. Find the radius of a sphere if the surface area of a hemisphere is 192� square centimeters.

6 cm

r


Surface Areas of Spheres

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ExercisesExercises

ExampleExample

Skills PracticeSurface Areas of Spheres

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In the figure, A is the center of the sphere, and plane Tintersects the sphere in circle E. Round to the nearesttenth if necessary.

1. If AE � 5 and DE � 12, find AD.

2. If AE � 7 and DE � 15, find AD.

3. If the radius of the sphere is 18 units and the radius of �E is 17 units, find AE.

4. If the radius of the sphere is 10 units and the radius of �E is 9 units, find AE.

5. If M is a point on �E and AD � 23, find AM.

Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

6. 7.

8. a hemisphere with a radius of the great circle 8 yards

9. a hemisphere with a radius of the great circle 2.5 millimeters

10. a sphere with the area of a great circle 28.6 inches

32 m7 in.

A

ED

T


In the figure, C is the center of the sphere, and plane Bintersects the sphere in circle R. Round to the nearesttenth if necessary.

1. If CR � 4 and SR � 14, find CS.

2. If CR � 7 and SR � 24, find CS.

3. If the radius of the sphere is 28 units and the radius of �R is 26 units, find CR.

4. If J is a point on �R and CS � 7.3, find CJ.

Find the surface area of each sphere or hemisphere. Round to the nearest tenth.

5. 6.

7. a sphere with the area of a great circle 29.8 meters

8. a hemisphere with a radius of the great circle 8.4 inches

9. a hemisphere with the circumference of a great circle 18 millimeters

10. SPORTS A standard size 5 soccer ball for ages 13 and older has a circumference of27–28 inches. Suppose Breck is on a team that plays with a 28-inch soccer ball. Find thesurface area of the ball.

89 ft6.5 cm

C

RS

B

Practice Surface Areas of Spheres

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Reading to Learn MathematicsSurface Areas of Spheres

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7

Pre-Activity How do manufacturers of sports equipment use the surface areas of spheres?


How would knowing the formula for the surface area of a sphere helpmanufacturers of beach balls?

Reading the Lesson1. In the figure, P is the center of the sphere. Name each of

the following in the figure.

a. three radii of the sphere

b. a diameter of the sphere

c. two chords of the sphere

d. a great circle of the sphere

e. a tangent to the sphere

f. the point of tangency

2. Determine whether each sentence is sometimes, always, or never true.

a. If a sphere and a plane intersect in more than one point, their intersection will be agreat circle.

b. A great circle has the same center as the sphere.c. The endpoints of a radius of a sphere are two points on the sphere.d. A chord of a sphere is a diameter of the sphere.e. A radius of a great circle is also a radius of the sphere.

3. Match each surface area formula with the name of the appropriate solid.

a. T � �r� � �r2 i. regular pyramid

b. T � Ph � 2B ii. hemisphere

c. T � 4�r2 iii. cylinder

d. T � �12�P� � B iv. prism

e. T � 2�rh � 2�r2 v. sphere

f. T � 3�r2 vi. cone

Helping You Remember4. Many students have trouble remembering all of the formulas they have learned in this

chapter. What is an easy way to remember the formula for the surface area of a sphere?

R S

T

P

W

V

X Y

U


Doubling SizesConsider what happens to surface area when the sides of a figure are doubled.

The sides of the large cube are twice the size of the sides of the small cube.

1. How long are the edges of the large cube?

2. What is the surface area of the small cube?

3. What is the surface area of the large cube?

4. The surface area of the large cube is how many times greater than that of the small cube?

The radius of the large sphere at the right is twice the radius of the small sphere.

5. What is the surface area of the small sphere?

6. What is the surface area of the large sphere?

7. The surface area of the large sphere is how many times greater than the surface area of the small sphere?

8. It appears that if the dimensions of a solid are doubled, the surface area is multiplied by ��.

Now consider how doubling the dimensions affects the volume of a cube.

The sides of the large cube are twice the size of the small cube.

9. How long are the edges of the large cube?

10. What is the volume of the small cube?

11. What is the volume of the large cube?

12. The volume of the large cube is how many times greater than that of the small cube?

The large sphere at the right has twice the radius of the small sphere.

13. What is the volume of the small sphere?

14. What is the volume of the large sphere?

15. The volume of the large sphere is how many times greater than the volume of the small sphere?

16. It appears that if the dimensions of a solid are doubled, the volume is multiplied by ��.

3 m

5 in.

3 m

5 in.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-712-7

Chapter 12 Test, Form 11212


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of eachquestion.

1. Which of these is part of an orthogonal drawing?A. a perspective view B. a corner viewC. a two-dimensional top view D. a three-dimensional view

For Questions 2–4, use the figure.

2. Identify this solid figure.A. square pyramid B. square prismC. triangular pyramid D. triangular prism

3. Name the base.A. �ABE B. �ABCD C. �CDE D. E

4. How many edges does this figure have?A. 3 B. 4 C. 6 D. 8

5. This net could be folded into a .A. tetrahedronB. square pyramidC. square prismD. triangular prism

6. Find the surface area of this solid.A. 9 in2 B. 27 in2

C. 36 in2 D. 54 in2

7. The areas of how many faces of a rectangular prism would be included in thelateral area?A. 2 B. 4 C. 6 D. 8

8. Find the surface area of a rectangular prism with a length of 8 inches, awidth of 5 inches, and a height of 2 inches.A. 15 in2 B. 66 in2 C. 80 in2 D. 132 in2

9. The area of each face of a cube is 60 square centimeters. Find the surfacearea of the cube.A. 120 cm2 B. 240 cm2 C. 360 cm2 D. 3600 cm2

10. The lateral area of a right cylinder with a radius of 10 feet is 320� squarefeet. Find the surface area of the cylinder.A. 220� ft2 B. 360� ft2 C. 420� ft2 D. 520� ft2

3 in.3 in.

3 in.

6 in.

4 in.4 in. 4 in.

4 in.

4 in.

?

E

AB

D

C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 12 Test, Form 1 (continued)1212

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

For Questions 11 and 12, use the figure.

11. Find the lateral area to the nearest tenth.A. 75.4 ft2 B. 62.8 ft2

C. 50.3 ft2 D. 25.1 ft2

12. Find the surface area to the nearest tenth.A. 75.4 ft2 B. 62.8 ft2 C. 50.3 ft2 D. 25.1 ft2

13. The lateral area of a regular pyramid is 300 square units. The perimeter of itsbase is 100 units. Find the slant height of the pyramid.A. 3 units B. 6 units C. 12 units D. 30 units


14. Find the lateral area.A. 108 cm2 B. 144 cm2

C. 162 cm2 D. 324 cm2

15. Find the surface area.A. 108 cm2 B. 144 cm2 C. 162 cm2 D. 324 cm2

16. Find the surface area to the nearest tenth.A. 546.6 units2 B. 989.6 units2

C. 1017.9 units2 D. 1046.2 units2

17. The radius of a right circular cone is 6 inches and the height is 8 inches. Findthe slant height.A. 2 in. B. 4 in. C. 10 in. D. 14 in.

18. A cone has a radius 17 inches long and slant height 20 inches long. Find thesurface area to the nearest tenth.A. 18,158.4 in2 B. 1976.1 in2 C. 1068.1 in2 D. 340 in2

19. Which could be the intersection of a sphere and a plane?A. line B. square C. oval D. point

20. A sphere has a diameter 42 centimeters long. Find the surface area to thenearest tenth.A. 5541.8 cm2 B. 2770.9 cm2 C. 2167.1 cm2 D. 527.8 cm2

Bonus Find the amount of glass needed to cover the sides of thegreenhouse shown. The bottom,front, and back are not glass.

15 ft

30 ft

9 ft

9 ft

8 ft8 ft

50

5

6

6 cm6 cm

9 cm

2 ft

4 ft

B:

NAME DATE PERIOD

Chapter 12 Test, Form 2A1212


Ass

essm

ents


1. What do the dark segments represent in an orthogonal drawing?A. a change in color B. where paper should be foldedC. a design on the surface D. a break in the surface

For Questions 2 and 3, use the figure.2. Identify the figure.

A. pyramid B. prismC. cone D. cylinder

3. Name the base.A. X B. Y C. X�Y� D. �Y

4. What name is given to a prism having five faces?A. pentagonal prism B. square prismC. triangular prism D. none of these

5. This net could be folded into a .A. cone B. cylinderC. sphere D. triangular prism

6. Find the surface area of the solid.A. 188 ft2 B. 240 ft2

C. 288 ft2 D. 480 ft2

7. The lateral area of a cube is 36 square inches. How long is each edge?A. �6� in. B. 3 in. C. 6 in. D. 9 in.

8. The lateral area of a prism is 56 square inches and the area of each base is 17 square inches. Find the surface area of the prism.A. 952 in2 B. 90 in2 C. 73 in2 D. 22 in2

9. Find the surface area of the outside of the open box.A. 1920 in2 B. 998 in2

C. 752 in2 D. 400 in2

10. The surface area of a right cylinder is 200� square centimeters and theradius is 4 centimeters. Find the height.A. 42 cm B. 25 cm C. 23 cm D. 21 cm

20 in.12 in.

8 in.

10 ft8 ft

6 ft

1

3

3

?

Y

X

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 12 Test, Form 2A (continued)1212

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

For Questions 11 and 12, use a right cylinder with a radius of 3 inchesand a height of 17 inches. Round to the nearest tenth.

11. Find the lateral area.A. 320.4 in2 B. 348.7 in2 C. 377.0 in2 D. 537.2 in2

12. Find the surface area.A. 320.4 in2 B. 348.7 in2 C. 377.0 in2 D. 537.2 in2

13. The lateral area of a regular pentagonal pyramid is 75 square inches and theslant height is 10 inches. Find the length of each side of the base.A. 15 in. B. 14 in. C. 7.5 in. D. 3 in.


14. Find the lateral area.A. 144 cm2 B. 144 � 24�3� cm2

C. 196 cm2 D. 288 cm2

15. Find the surface area.A. 144 cm2 B. 144 � 24�3� cm2 C. 196 cm2 D. 288 cm2

For Questions 16 and 17, use the figure.Round to the nearest tenth.

16. Find the lateral area.A. 44.0 in2 B. 75.4 in2 C. 88.0 in2 D. 100.5 in2

17. Find the surface area.A. 44.0 in2 B. 75.4 in2 C. 88.0 in2 D. 100.5 in2

18. Find the surface area of this model rocket to the nearest tenth.A. 2890.3 cm2 B. 2576.1 cm2

C. 2513.3 cm2 D. 2261.9 cm2


19. Identify a chord.A. E�F� B. �B C. B�D� D. AD�

20. Find the surface area to the nearest tenth.A. 4536.5 m2 B. 2268.2 m2 C. 477.5 m2 D. 238.8 m2

Bonus Find the surface area of the figure to the nearest tenth.

4 ft

12 ft

1 ft

19 mA C

D

E F

B

10 cm

30 cm

12 cm

12 in.2 in.

12 in.

4 in.

B:

NAME DATE PERIOD

Chapter 12 Test, Form 2B1212


Ass

essm

ents


1. Given the corner view of a figure, which is the top view?A. B. C. D.


2. Identify the figure.A. pyramid B. prismC. cone D. cylinder

3. Name a base.A. �M B. N C. M�N� D. M

4. What name is given to a pyramid having seven faces?A. heptagonal pyramid B. hexagonal pyramidC. pentagonal pyramid D. octagonal pyramid

5. This net could be folded into a .A. rectangular pyramidB. rectangular prismC. triangular pyramidD. triangular prism

6. Find the surface area of the solid.A. 88 cm2 B. 102 cm2

C. 156 cm2 D. 160 cm2

7. Find the lateral area of an equilateral triangular prism if the area of eachlateral face is 10 square centimeters.A. 10�3� cm2 B. 30 cm2 C. 50 cm2 D. 100 cm2

8. The surface area of a cube is 96 square inches. Find the length of an edge.A. �24� in. B. 4 in. C. 8 in. D. 16 in.

9. The surface area of a rectangular prism is 190 square inches, the length is 10 inches, and the width 3 inches. Find the height.A. 30 in. B. 20 in. C. 10 in. D. 5 in.

10. A right cylinder has a radius of 2 feet and a height of 5 feet. Find its surfacearea.A. 20� ft2 B. 28� ft2 C. 36� ft2 D. 40� ft2

4 cm

5 cm

2 cm

8 cm

4 in.

8 in.

8 in.8 in.

5 in.

4 in.

5 in. 5 in.

?

N

M

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 12 Test, Form 2B (continued)1212

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

For Questions 11 and 12, use a right cylinder with a radius of 5 centimeters and a height of 22 centimeters. Round to the nearest tenth.

11. Find the lateral area.A. 848.2 cm2 B. 769.7 cm2 C. 691.2 cm2 D. 345.6 cm2

12. Find the surface area.A. 848.2 cm2 B. 769.7 cm2 C. 691.2 cm2 D. 345.6 cm2

13. Find the lateral area of the conical hat to the nearest tenth.A. 942.5 in2 B. 408.4 in2

C. 204.2 in2 D. 188.5 in2

For Questions 14 and 15, use a tetrahedron that has edges of length 12 centimeters.14. Find the lateral area.

A. 48�3� cm2 B. 96�3� cm2 C. 108�3� cm2 D. 144�3� cm2

15. Find the surface area.A. 48�3� cm2 B. 96�3� cm2 C. 108�3� cm2 D. 144�3� cm2

For Questions 16 and 17, use the figure. Round to the nearest tenth.16. Find the lateral area.

A. 75.4 cm2 B. 103.7 cm2

C. 131.9 cm2 D. 150.8 cm2

17. Find the surface area.A. 75.4 cm2 B. 103.7 cm2 C. 131.9 cm2 D. 150.8 cm2

18. Find the surface area of the solid to the nearest tenth.A. 62.8 cm2 B. 56.5 cm2

C. 47.1 cm2 D. 37.7 cm2

19. Name a tangent to the sphere.A. Y�U� B. X�Z�C. UZ� D. WV��

20. The surface area of a sphere is 64� square centimeters. Find the radius.A. 16 cm B. 8 cm C. 4 cm D. 2 cm

Bonus Find the surface area of the frustum of a square pyramid.

3 ft

2 ft

4 ft

X Z

U

VW

Y

1 cm

6 cm

3 cm3 cm

3 cm8 cm

12 in.

5 in.

B:

NAME DATE PERIOD

Chapter 12 Test, Form 2C1212


Ass

essm

ents

1. Given the corner view of a figure,sketch the front view.

2. Name the faces of the solid.

3. Identify the solid.

4. How many faces does a dodecahedron have?

5. Draw a net for the solid.

6. Find the surface area of the solid.

7. Find the lateral area of a triangular prism with a height of 8 centimeters, and with bases having sides that measure 4 centimeters, 5 centimeters, and 6 centimeters.

8. Find the surface area of the prism.

9. Find the surface area of the solid to the nearest tenth.

10. A gallon of paint costs $12.99 and covers 400 square feet. Howmany gallons are needed to paint two coats on the walls andceiling (not the floor) of a rectangular room that is 30 feet long,18 feet wide, and 8 feet high? Round to the next whole gallon.

1 in.

1 in.

1 in.

0.25 in.

25

25

5014

12 in.6 in.

5 in.

65

4

P S

RQ

T

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

6 6

5 4 44

4

5

NAME DATE PERIOD

SCORE


Chapter 12 Test, Form 2C (continued)1212

11. Find the lateral area of a right cylinder with a diameter of 8.6 yards and a height of 19.4 yards. Round to the nearest tenth.

12. The surface area of a cylinder is 180� square inches and theheight is 9 inches. Find the radius.

For Questions 13 and 14, use a regular hexagonal pyramidwith base edges of 10 inches and a slant height of 9 inches.


14. Find the surface area.

15. Find the lateral area of the triangular pyramid.

16. The surface area of a regular pyramid is 276 square centimeters,the slant height measures 8 centimeters, and the area of thebase is 50 square centimeters. Find the perimeter of the base.

For Questions 17 and 18, use a right circular cone with aradius of 4 feet and a height of 3 feet. Round to the nearesttenth.



19. Plane A intersects sphere R in �X.Find the radius of the sphere.

20. Find the surface area of this hemisphere to the nearest tenth.

Bonus The surface area of the exterior of a hollow rubber ball is

16� square inches. The rubber is �14� inch thick. Find the

surface area of the interior of the ball.

5 cm

R

XA

15 m 8 m

101010

12

12

12

NAME DATE PERIOD

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 12 Test, Form 2D1212


Ass

essm

ents

1. Given the corner view of a figure,sketch the back view.

2. Name the edges of the solid.


4. How many edges does a cube have?



7. Find the lateral area of a regular pentagonal prism if theperimeter of the base is 50 inches and the height is 15 inches.


9. A gallon of paint costs $18 and covers 300 square feet. Howmany gallons are needed to paint the walls and bottom of arectangular swimming pool whose length is 25 feet, width is16 feet, and depth is 4 feet? Round to the next whole gallon.


6 in.6 in.

4 in.

4 in.

9 ft5 ft

6 ft

17 cm

17 cm16 cm

12 cm

88

8

6

6

6

G

I

H

J

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

88

8 8

88

6

66

NAME DATE PERIOD

SCORE


Chapter 12 Test, Form 2D (continued)1212

11. A right cylinder has a diameter of 23.6 meters and a height of11.4 meters. Find the lateral area to the nearest tenth.

12. The surface area of a right cylinder is 252� square feet andthe height is 11 feet. Find the radius.

For Questions 13 and 14, use a regular octagonal pyramidwith base edges 9 feet long, slant height 15 feet, and a basewith an apothem of 10.86 feet.


14. Find the surface area to the nearest tenth.

15. Find the lateral area of the solid to the nearest tenth.

16. The surface area of a regular decagonal pyramid is 1800 squarefeet, the area of the base is 120 square feet, and the slantheight is 15 feet. Find the length of each side of the base.

For Questions 17 and 18, use a cone with a radius of 5 centimeters and a height of 12 centimeters. Round to thenearest tenth.



19. Plane A intersects sphere B in �C.Find BC.

20. Find the surface area of this hemisphere to the nearest tenth.

Bonus The length of each side of a cube is 6 inches long. Find thesurface area of a sphere inscribed in the cube.

11 in.

B

CA

13 m12 m

1818

10

NAME DATE PERIOD

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 12 Test, Form 31212


Ass

essm

ents

1. Draw the back view of a figure given its orthogonaldrawing.

2. Name the bases of the solid.


4. How many faces does an icosahedron have?



7. Find the lateral area of a triangular prism with a righttriangular base with legs that measure 2 feet and 3 feet and a height of 7 feet.


For Questions 9 and 10, use a right cylinder with adiameter of 96.4 feet and a height of 58.9 feet. Round to thenearest tenth.



11. If the height of a right cylinder is multiplied by four and theradius remains the same, what happens to the lateral area?

6

10

20

84

3

��21

3

2

2

2

5

XZ

YV

U

W


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

52

2

2

22

2

2

NAME DATE PERIOD

SCORE


Chapter 12 Test, Form 3 (continued)1212

For Questions 12 and 13, use the solid.



14. If the length of each side of a cube is tripled, what happens tothe surface area?

For Questions 15 and 16, use a right circular cone with aradius of 7 inches and a height of 8 inches. Round to thenearest tenth.



17. Find the surface area of this frustum of a cone, to the nearest tenth.

18. Parallel planes A and B intersect sphere C in circles D and E and DC � 6. Find CE.

19. Write a formula for the surface area of a hemisphere in termsof � and the radius r.

20. Find the surface area of the solid to the nearest square foot.

Bonus Find the surface area of the solid to the nearest square foot. Do notinclude the area of the base.

8 ft

10 ft

6 ft

8 ft 22 ft

B

E

C

D8

4

A

B

D

6 cm

4 cm

2 cm

13 in.

12 in.

NAME DATE PERIOD

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 12 Open-Ended Assessment1212


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. a. Complete this chart.

b. Write a formula relating the number of edges to the number of facesand vertices.

2. Explain the difference between the lateral area and the surface area of a prism.

3. a. Draw and label a pyramid.

b. Name the base.

c. Name the lateral faces.

d. Draw a net for your pyramid.

4. Draw an oblique cylinder and a right cylinder.

5. a. Draw and label the dimensions of a solid figure composed of three ormore different solids studied in this chapter.

b. Find the surface area.

6. Write a practical application problem involving the surface area or lateralarea of a solid figure studied in this chapter.

Figure No. of Edges (e) No. of Faces (f ) No. of Vertices (v ) f � v

Triangular Pyramid 6 4 4 8

Triangular Prism

Cube

Square Pyramid

Hexagonal Prism

Hexagonal Pyramid

NAME DATE PERIOD

SCORE


Chapter 12 Vocabulary Test/Review1212

Choose from the terms above to complete each sentence.

1. The height of each lateral face of a regular pyramid is calleda(n) .

2. A(n) has a circular base and a vertex.

3. A(n) is a set of points in space that are a givendistance from a given point.

4. The view of a solid figure from the corner is called a cornerview or .

5. The five types of regular polyhedra are called the .

6. A(n) is a polyhedron with two parallel congruent facescalled bases.

7. A polyhedron that has all but one face intersecting at onepoint is a(n) .

8. The is the sum of the areas of all the faces of the solid.

9. If the axis of a cylinder is also the altitude, then the cylinderis called a(n) .

10. A sphere is separated by a great circle into two congruenthalves, each called a(n) .

Define each term.

11. lateral area

12. right prism

13. great circle

?

?

?

?

?

?

?

?

?

?1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

axisbasescircular coneconecorner viewcross sectioncylinderedgesface

great circlehemispherelateral arealateral edgeslateral facesnetoblique coneoblique cylinder

oblique prismorthogonal drawingperspective viewPlatonic solidspolyhedronprismpyramidregular polyhedron

regular prismregular pyramidright coneright cylinderright prismslant heightspheresurface area

NAME DATE PERIOD

SCORE

Chapter 12 Quiz (Lessons 12–1 and 12–2)

1212


Ass

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ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

1. Given the corner view of a figure,draw the left view.

2. Draw a rectangular prism.

3. Draw a net for this solid.

4. STANDARDIZED TEST PRACTICE Find the surface area of the solid formed by folding this net.A. 48 in2 B. 64 in2

C. 72 in2 D. 88 in2 2 in.

2 in.

2 in.

4 in.

4 in.

6 in.

210

1022

22

2

22 22

2

2222

2


1212

1.

2.

3.

4.

5.

1. The lateral area of a prism is 70 square inches and theperimeter of its base is 35 inches. Find the height.

2. The surface area of a rectangular prism is 592 square units.The height is 10 units and the width is 8 units. Find the length.

3. Describe the shape of a net that could be used to calculate thelateral area of a cylinder.

For Questions 4 and 5, use the solid figure. Round to the nearest tenth.



11

56

NAME DATE PERIOD

SCORE



1212

1.

2.

3.

4.

5.

For Questions 1 and 2, use the solid figure.



3. The lateral area of a pyramid is 200 square centimeters and thebase is a rectangle with a length measuring 15 centimetersand a width measuring 4 centimeters. Find the surface area ofthe pyramid.

For Questions 4 and 5, use a right circular cone with aradius of 5 feet and a slant height of 12 feet. Round to thenearest tenth.



6

9

9

9

NAME DATE PERIOD

SCORE

Chapter 12 Quiz (Lesson 12–7)

1212

1.

2.

3.

4.

5.

For Questions 1–3, use the figure.

1. Name a chord of sphere D that is not adiameter.

2. Name a tangent to the sphere.

3. Name a great circle.

4. A sphere has a radius that is 18 inches long. Find the surfacearea to the nearest tenth.

5. The radius of a sphere is doubled. How is the surface areachanged?

E

A

H

F

G

BC

D

NAME DATE PERIOD

SCORE

Chapter 12 Mid-Chapter Test (Lessons 12–1 through 12–3)

1212


Ass

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ents

1. Which of these are Platonic solids?A. cylinder, cone, sphere B. prism, pyramidC. tetrahedron, octahedron, dodecahedron D. hexagon, octagon, dodecagon

2. Which of the following describes a tetrahedron?A. triangular pyramid B. triangular prismC. square pyramid D. cone

3. The surface area of a prism is 120 square centimeters and the area of eachbase is 32 square centimeters. Find the lateral area of the prism.A. 184 cm2 B. 152 cm2 C. 86 cm2 D. 56 cm2

For Questions 4 and 5, use the solid figure. Round to the nearest tenth.4. Find the lateral area.

A. 9289.1 ft2 B. 9434.2 ft2

C. 10,965.4 ft2 D. 12,641.8 ft2

5. Find the surface area.A. 9289.1 ft2 B. 9434.2 ft2 C. 10,965.4 ft2 D. 12,641.8 ft2

46.2 ft

64 ft

6.

7.

8.

9.

10.

6

5

5

4

4

4 53

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

6. Draw the top view of the solid figure.

7. Draw a net of the solid.

8. Find the surface area of the solid formed by folding this net.

9. Find the surface area of the regular hexagonal prism to the nearest tenth.

10. Find the total surface area to the nearest tenth. 4 in.

6 in.

1 in.

2 6

30

15

15

18

5

64

3

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 12 Cumulative Review(Chapters 1–12)

1212

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

1. Name two obtuse vertical angles.(Lesson 1-5)

2. Use the statements p: �8 � 3 � 5 and q: A triangle has foursides to find the truth value of p � q. (Lesson 2-2)

3. Find the value of �1x�, where x initially equals 2. Then, use that

value as the next x in the expression. Repeat the process anddescribe your observations. (Lesson 6-6)

4. Find m�X and m�Z to the nearest degree and y to the nearest centimeter.(Lesson 7-7)

5. Each side of a rhombus is 18 inches long. One diagonal makesa 40° angle with one side. What is the length of each diagonalto the nearest tenth? (Lesson 8-5)

6. Find the magnitude and direction of ST� for S(4, �1) andT(�6, 5). Round to the nearest tenth. (Lesson 9-6)

7. A certain figure is the locus of all points in a plane equidistantfrom point B. What is the shape of this figure? (Lesson 10-1)

8. Find m�C. (Lesson 10-6)

9. Find the area and perimeter.(Lesson 11-1)

10. The surface area of a cube is 1261.5 square meters. Find thelength of a lateral edge of the cube. (Lesson 12-3)

11. The surface area of a right cylinder is 502.7 square inches andthe height is 11 inches. Find the radius of the base to thenearest tenth. (Lesson 12-4)

6 cm

4 cm

4 cm

7 cm

3 cm2 cm

88� 32�

E

A

D

BC

73�

32 cm

25 cmy

Z Y

X

18�

74�

EA D

BF

C

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–12)


1. Which method could you use to prove B�E� � A�C� if AF � BF? (Lesson 4-5)

A. Show that �ABE � �BAC by SSS,then B�E� � A�C� by CPCTC.

B. Show that �ABE � �BAC by ASA, then B�E� � A�C� by CPCTC.C. Show that �BFE � �AFC by SAS, then B�E� � A�C� by CPCTC.D. Show that �ABE � �BAC by AAS, then B�E� � A�C� by CPCTC.

2. Find a. (Lesson 6-3)

E. 28.5 F. 6.3G. 12.6 H. 14

3. Find r. (Lesson 7-6)

A. about 34.0 B. about 11.8C. about 8.9 D. about 6.6

4. A square has side length 18 centimeters. Find the area of thesquare. (Lesson 8-5)

E. 36 cm2 F. 40 cm2 G. 81 cm2 H. 324 cm2

5. What can you assume from the figure? (Lesson 10-3)

A. �ABC is isosceles.B. �ABC is equilateral.C. DF � EGD. radius of �O � x � y

6. Points D, E, and F are on a circle so that mDEF�� 210. Suppose

point G is randomly located on the same circle so that it does notcoincide with D, E, or F. What is the probability that m�DGF �105? (Lesson 10-4)

E. �172� F. �1

52� G. 1 H. �

34�

7. Which net could be folded into a triangular prism? (Lesson 12-2)

A. B. C. D.

8. Find the surface area of a square pyramid with a height of 9 centimeters and base with a side measuring 24 centimeters.(Lesson 12-5)

E. 1296 cm2 F. 1806 cm2 G. 2016 cm2 H. 8640 cm2

x

yxO

B

A F G

C

E

D

97�

49�

8.9

r

Q R

P

40�

40�

4.2

12.6

9.5

a

A

B E

C

F

NAME DATE PERIOD

SCORE 1212

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7.

8. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents


Standardized Test Practice (continued)

9. Quadrilateral PQSRis a rectangle.Find a. (Lesson 8-4)

10. Find x. Assume that segments that appeartangent are tangent.(Lesson 10-5)

2x � 8

5x � 7

M

N

(2a � 1)�(6a � 21)�

P Q

SR

NAME DATE PERIOD

1212

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short Response

Instructions: Show your work or explain in words how you found your answer.

9. 10.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

11. Find the length of SR� to the nearest tenth. (Lesson 10-2)

12. Find x. (Lesson 10-7)

13. Find the area of the shaded region to the nearest tenth. (Lesson 11-3)

14. Identify the solid. (Lesson 12-1)

15. A right circular cone has a slant height of 15 inches and aradius that is 25 inches long. Find the surface area to thenearest tenth. (Lesson 12-6)

16. A ball has a diameter of 26.5 centimeters. Find the surfacearea of the ball to the nearest tenth. (Lesson 12.7)

10 cm

60� 60�

60�

15

3018

x

50�

3 mT

S

R

11.

12.

13.

14.

15.

16.

1 4 5

Standardized Test PracticeStudent Record Sheet (Use with pages 684–685 of the Student Edition.)

1212

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 9 DCBADCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

Also enter your answer by writing each number or symbol in a box. Then fill inthe corresponding oval for that number or symbol.

10 (grid in) 10 11 12

11 (grid in)

12 (grid in)0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

Record your answers for Questions 13–14 on the back of this paper.


Stu

dy G

uid

e a

nd I

nte

rven

tion

Th

ree-

Dim

ensi

on

al F

igu

res

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1

©G

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etry

Lesson 12-1

Dra

win

gs

of

Thre

e-D

imen

sio

nal

Fig

ure

sT

o w

ork

wit

h a

th

ree-

dim

ensi

onal

obje

ct,a

use

ful

skil

l is

th

e ab

ilit

y to

mak

e an

ort

hog

onal

dra

win

g,w

hic

h i

s a

set

of t

wo-

dim

ensi

onal

dra

win

gs o

f th

e di

ffer

ent

side

s of

th

e ob

ject

.For

a s

quar

e py

ram

id,y

ou w

ould

show

th

e to

p vi

ew,t

he

left

vie

w,t

he

fron

t vi

ew,a

nd

the

righ

t vi

ew.

Dra

w t

he

bac

k v

iew

of

the

figu

re

give

n t

he

orth

ogon

al d

raw

ing.

•T

he

top

view

in

dica

tes

two

colu

mn

s.•

Th

e le

ft v

iew

in

dica

tes

that

th

e h

eigh

t of

fig

ure

is

thre

e bl

ocks

.•

Th

e fr

ont

view

in

dica

tes

that

th

e co

lum

ns

hav

e h

eigh

ts 2

an

d 3

bloc

ks.

•T

he

righ

t vi

ew i

ndi

cate

s th

at t

he

hei

ght

of t

he

figu

re i

s th

ree

bloc

ks.

Use

blo

cks

to m

ake

a m

odel

of

the

obje

ct.T

hen

use

you

r m

odel

to

draw

th

e ba

ck v

iew

.Th

e ba

ck v

iew

in

dica

tes

that

th

e co

lum

ns

hav

e h

eigh

ts

3 an

d 2

bloc

ks.

Dra

w t

he

bac

k v

iew

an

d c

orn

er v

iew

of

a fi

gure

giv

en e

ach

ort

hog

onal

dra

win

g.

1.2.

3.4.

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

back

vie

wob

ject

top

view

left

view

front

vie

wrig

ht v

iew

obje

ctto

p vi

ewle

ft vi

ewfro

nt v

iew

right

vie

w

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

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etry

Iden

tify

Th

ree-

Dim

ensi

on

al F

igu

res

A p

olyh

edro

nis

a s

olid

wit

h a

ll f

lat

surf

aces

.Eac

h s

urf

ace

of a

pol

yhed

ron

is

call

ed a

fac

e,an

d ea

ch l

ine

segm

ent

wh

ere

face

sin

ters

ect

is c

alle

d an

ed

ge.T

wo

spec

ial

kin

ds o

f po

lyh

edra

are

pri

sms,

for

wh

ich

tw

o fa

ces

are

con

gru

ent,

para

llel

bas

es,a

nd

pyr

amid

s,fo

r w

hic

h o

ne

face

is

a ba

se a

nd

all

the

oth

erfa

ces

mee

t at

a p

oin

t ca

lled

th

e ve

rtex

.Pri

sms

and

pyra

mid

s ar

e n

amed

for

th

e sh

ape

ofth

eir

base

s,an

d a

regu

lar

poly

hed

ron

has

a r

egu

lar

poly

gon

as

its

base

.

Oth

er s

olid

s ar

e a

cyli

nd

er,w

hic

h h

as c

ongr

uen

t ci

rcu

lar

base

s in

par

alle

l pl

anes

,a c

one,

wh

ich

has

on

e ci

rcu

lar

base

an

d a

vert

ex,a

nd

a sp

her

e.

Iden

tify

eac

h s

olid

.Nam

e th

e b

ases

,fac

es,e

dge

s,an

d v

erti

ces.

pent

agon

alpr

ism

squa

repy

ram

idpe

ntag

onal

pyra

mid

rect

angu

lar

pris

msp

here

cone

cylin

der

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Th

ree-

Dim

ensi

on

al F

igu

res

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1

Exam

ple

Exam

ple

a.

Th

e fi

gure

is

a re

ctan

gula

r py

ram

id.T

he

base

is

rect

angl

e A

BC

D,a

nd

the

fou

r fa

ces

�A

BE

,�B

CE

,�

CD

E,a

nd

�A

DE

mee

t at

ver

tex

E.T

he

edge

s ar

eA �

B�,B�

C�,C�

D�,A�

D�,A�

E�,B�

E�,C�

E�,a

nd

D�E�

.Th

e ve

rtic

esar

e A

,B,C

,D,a

nd

E.

E

AB

CD

b.

Th

is s

olid

is

a cy

lin

der.

Th

etw

o ba

ses

are

�O

and

�P

.

PO

Exer

cises

Exer

cises

Iden

tify

eac

h s

olid

.Nam

e th

e b

ases

,fac

es,e

dge

s,an

d v

erti

ces.

1.2.

3.4.

Th

e fa

ces

are

pen

tag

on

s P

WX

YZ

�D

EF

.Th

e fa

ces

are

�A

BC

and

an

d R

QV

US

and

par

alle

log

ram

s �

DE

Fan

d p

aral

lelo

gra

ms

AC

FD

,Z

SU

Y,X

YU

V,W

XV

Q,P

RQ

W,a

nd

A

BE

D,a

nd

BE

FC

.Th

e ed

ges

are

ZP

RS

.Th

e ed

ges

are

Z�S�

,Y�U�

,X�V�

,W�Q�

,A�

B�,B�

C�,A�

C�,C�

F�,A�

D�,B�

E�,D�

E�,E�

F�,P�

R�,Z�

Y�,Y�

X�,X�

W�,W�

P�,P�

Z�,S�

U�,U�

V�,V�

Q�,

and

D�F�.

Th

e ve

rtic

es a

re A

,B,C

,Q�

R�,a

nd

R�S�

.Th

e ve

rtic

es a

re P

,W,X

,D

,E,a

nd

F.

Y,Z

,R,Q

,V,U

,an

d S

.

Th

e so

lid is

atr

ian

gu

lar

pri

sm.

Th

e b

ases

are

�A

BC

and

AB

CF

DE

Th

e fi

gu

re is

ap

enta

go

nal

pri

sm.T

he

bas

es a

re p

enta

go

ns

PW

XY

Zan

d R

QV

US

.P

WQV

ZX

U

S R

Y

Th

e so

lid is

a s

ph

ere

wit

h c

ente

r T

.T

Th

e so

lid is

a c

on

e.It

has

�S

fo

r a

bas

ean

d v

erte

x R

.SR

Answers (Lesson 12-1)


An

swer

s

Skil

ls P

ract

ice

Th

ree-

Dim

ensi

on

al F

igu

res

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1

©G

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etry

Lesson 12-1

Dra

w t

he

bac

k v

iew

an

d c

orn

er v

iew

of

a fi

gure

giv

en e

ach

ort

hog

onal

dra

win

g.

1.2.

Iden

tify

eac

h s

olid

.Nam

e th

e b

ases

,fac

es,e

dge

s,an

d v

erti

ces.

3.

rect

ang

ula

r p

rism

;sa

mp

le a

nsw

er:

bas

es:

�R

ST

U,�

VW

XY

;fa

ces:

�R

ST

U,�

RS

XY

,�V

WX

Y,�

TU

VW

,�S

TW

X,�

RU

VY

;ed

ges

:R�

S�,R�

U�,S�

T�,T�U�

,S�X�

,R�Y�

,T�W�

,U�V�

,V�Y�

,W�X�

,X�Y�

,V�W�

;ve

rtic

es:

R,S

,T,U

,V,W

,X,a

nd

Y

4.

pen

tag

on

al p

yram

id;

bas

e:p

enta

go

n A

BC

DE

;fa

ces:

pen

tag

on

AB

CD

E,�

AF

B,�

BF

C,�

CF

D,�

DF

E,�

AF

E;

edg

es:

A�B�

,B�C�

,C�D�

,D�E�

,A�E�

,A�F�,

B�F�,

C�F�,

D�F�,

E�F�

;ve

rtic

es:

A,B

,C,D

,E,a

nd

F

5.

con

e;b

ase:

circ

le R

;ve

rtex

:S

SR

F

AB

CD

E

UT

WV

RS

XY

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

©G

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etry

Dra

w t

he

bac

k v

iew

an

d c

orn

er v

iew

of

a fi

gure

giv

en e

ach

ort

hog

onal

dra

win

g.

1.2.

Iden

tify

eac

h s

olid

.Nam

e th

e b

ases

,fac

es,e

dge

s,an

d v

erti

ces.

3.

trap

ezo

idal

pri

sm;

bas

es:

trap

ezo

id H

IJK

,tra

pez

oid

LM

NO

;fa

ces:

trap

ezo

id H

IJK

,tra

pez

oid

LM

NO

,�H

KLO

,�H

INO

,�IJ

MN

,�JK

LM;

edg

es:H�

I�,I�J�

,J�K�

,H�K�

,H�O�

,O�N�

,I�N�

,K�L�,

L�O�,L�

M�,J�

M�,M�

N�;

vert

ices

:H,I

,J,K

,L,M

,N,a

nd

O

4.

cylin

der

;b

ases

:ci

rcle

T,c

ircl

e Z

5.M

INER

ALS

Pyr

ite,

also

kn

own

as

fool

’s g

old,

can

for

m c

ryst

als

that

are

per

fect

cu

bes.

Su

ppos

e a

gem

olog

ist

wan

ts t

o cu

t a

cube

of

pyri

te t

o ge

t a

squ

are

and

a re

ctan

gula

rfa

ce.W

hat

cu

ts s

hou

ld b

e m

ade

to g

et e

ach

of

the

shap

es?

Illu

stra

te y

our

answ

ers.

a cu

t p

aral

lel t

o t

he

bas

es t

o

a cu

t th

rou

gh

dia

go

nal

ly o

pp

osi

te t

op

g

et a

sq

uar

ean

d b

ott

om

ed

ges

to

get

a r

ecta

ng

le

Z T

IJ

MH

K

NOL

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

back

vie

wco

rner

vie

w

top

view

left

view

front

vie

wrig

ht v

iew

Pra

ctic

e (

Ave

rag

e)

Th

ree-

Dim

ensi

on

al F

igu

res

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1



Readin

g t

o L

earn

Math

em

ati

csT

hre

e-D

imen

sio

nal

Fig

ure

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-1

12-1

©G

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etry

Lesson 12-1

Pre-

Act

ivit

yW

hy

are

dra

win

gs o

f th

ree-

dim

ensi

onal

str

uct

ure

s va

luab

le t

oar

cheo

logi

sts?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-1 a

t th

e to

p of

pag

e 63

6 in

you

r te

xtbo

ok.

Wh

y do

you

th

ink

arch

eolo

gist

s w

ould

wan

t to

kn

ow t

he

deta

ils

of t

he

size

san

d sh

apes

of

anci

ent

thre

e-di

men

sion

al s

tru

ctu

res?

Sam

ple

an

swer

:T

his

info

rmat

ion

mig

ht

pro

vid

e cl

ues

ab

ou

t co

nst

ruct

ion

met

ho

ds

and

to

ols

an

d h

elp

arc

heo

log

ists

tra

ce a

dva

nce

s in

con

stru

ctio

n m

eth

od

s ov

er t

ime.

Rea

din

g t

he

Less

on

1.M

atch

eac

h d

escr

ipti

on f

rom

th

e fi

rst

colu

mn

wit

h o

ne

of t

he

term

s fr

om t

he

seco

nd

colu

mn

.(S

ome

of t

he

term

s m

ay b

e u

sed

mor

e th

an o

nce

or

not

at

all.)

a.a

poly

hed

ron

wit

h t

wo

para

llel

con

gru

ent

base

sv

b.

the

set

of p

oin

ts i

n s

pace

th

at a

re a

giv

en d

ista

nce

fro

m a

give

n p

oin

tvi

iic.

a re

gula

r po

lyh

edro

n w

ith

eig

ht

face

si

d.

a po

lyh

edro

n t

hat

has

all

fac

es b

ut

one

inte

rsec

tin

g at

on

epo

int

xie.

a li

ne

segm

ent

wh

ere

two

face

s of

a p

olyh

edro

n i

nte

rsec

tiv

f.a

soli

d w

ith

con

gru

ent

circ

ula

r ba

ses

in a

pai

r of

par

alle

lpl

anes

vii

g.a

regu

lar

poly

hed

ron

wh

ose

face

s ar

e sq

uar

esx

h.

a fl

at s

urf

ace

of a

pol

yhed

ron

ii

i.oc

tah

edro

nii

.fa

ceii

i.ic

osah

edro

niv

.ed

gev.

pris

mvi

.do

deca

hed

ron

vii.

cyli

nde

rvi

ii.s

pher

eix

.co

ne

x.h

exah

edro

nxi

.py

ram

idxi

i.te

trah

edro

n

2.F

ill

in t

he

mis

sin

g n

um

bers

,wor

ds,o

r ph

rase

s to

com

plet

e ea

ch s

ente

nce

.

a.A

tri

angu

lar

pris

m h

as

vert

ices

.It

has

fa

ces:

base

s ar

e co

ngr

uen

t

,an

d fa

ces

are

para

llel

ogra

ms.

b.

A r

egu

lar

octa

hed

ron

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.

8 in

.4

m

3 m

10 m

pent

agon

al p

rismal

titud

e

late

ral

edge

late

ral

face

Exam

ple

Exam

ple

Exer

cises

Exer

cises

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Surf

ace

Are

as o

f Pr

ism

sT

he

surf

ace

area

of

a pr

ism

is

the

late

ral

area

of

the

pris

m p

lus

the

area

s of

th

e ba

ses.

Su

rfac

e A

rea

If th

e to

tal s

urfa

ce a

rea

of a

pris

m is

Tsq

uare

uni

ts,

its h

eigh

t is

of

a P

rism

hun

its,

and

each

bas

e ha

s an

are

a of

Bsq

uare

uni

ts a

nd a

pe

rimet

er o

f P

units

, th

en T

�L

�2B

.

Fin

d t

he

surf

ace

area

of

the

tria

ngu

lar

pri

sm a

bov

e.F

ind

the

late

ral

area

of

the

pris

m.

L�

Ph

Late

ral a

rea

of a

pris

m

�(1

8)(1

0)P

�18

, h

�10

�18

0 cm

2M

ultip

ly.

Fin

d th

e ar

ea o

f ea

ch b

ase.

Use

th

e P

yth

agor

ean

Th

eore

m t

o fi

nd

the

hei

ght

of t

he

tria

ngu

lar

base

.

h2

�32

�62

Pyt

hago

rean

The

orem

h2

�27

Sim

plify

.

h�

3�3�

Take

the

squ

are

root

of

each

sid

e.

B�

�1 2��

base

�h

eigh

tA

rea

of a

tria

ngle

��1 2� (

6)(3

�3�)

or 1

5.6

cm2

Th

e to

tal

area

is

the

late

ral

area

plu

s th

e ar

ea o

f th

e tw

o ba

ses.

T�

180

�2(

15.6

)S

ubst

itutio

n

�21

1.2

cm2

Sim

plify

.

Fin

d t

he

surf

ace

area

of

each

pri

sm.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

3.

1024

in2

84 m

232

4 m

2

4.5.

6.

619.

1 in

241

5.4

in2

384

m2

8 m

8 m

8 m

8 in

.

8 in

.

8 in

.

15 in

.6

in.

12 in

.

12 m

5 m

6 m

3 m

6 m

4 m

24 in

.

10 in

.

8 in

.

6 cm

6 cm

6 cm

10 c

mh

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Su

rfac

e A

reas

of

Pri

sms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Su

rfac

e A

reas

of

Pri

sms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3

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etry

Lesson 12-3

Fin

d t

he

late

ral

area

of

each

pri

sm.

1.2.

480

un

its2

(sq

uar

e b

ase)

240

un

its2

(8 �

12 b

ase)

528

un

its2

(rec

tan

gu

lar

bas

e)28

8 u

nit

s2(1

2 �

6 b

ase)

336

un

its2

(8 �

6 b

ase)

3.4.

120

un

its2

324

un

its2

Fin

d t

he

surf

ace

area

of

each

pri

sm.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

5.6.

600

un

its2

782

un

its2

7.8.

312.

2 u

nit

s285

.2 u

nit

s27

4

39

610

15

7

13

18

8

6

9

9

9

12

1056

8

8

6

12

12

12

10

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eom

etry

Fin

d t

he

late

ral

area

of

each

pri

sm.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

1920

un

its2

(sq

uar

e b

ase)

224.

3 u

nit

s214

10 u

nit

s2(r

ecta

ng

ula

r b

ase)

3.4.

132

un

its2

123.

5 u

nit

s2

Fin

d t

he

surf

ace

area

of

each

pri

sm.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

5.6.

514

un

its2

311.

2 u

nit

s2

7.8.

263.

7 u

nit

s216

80 u

nit

s2

9.C

RA

FTS

Bec

ca m

ade

a re

ctan

gula

r je

wel

ry b

ox in

her

art

cla

ss a

nd p

lans

to

cove

r it

in r

ed

silk

.If

the

jew

elry

box

is

6�1 2�

inch

es l

ong,

4 �1 2�

inch

es w

ide,

and

3 in

ches

hig

h,f

ind

the

surf

ace

area

th

at w

ill

be c

over

ed.

124�

1 2�in

2

14

14

23

9

95

6

13

79

4

17

4

4

9.5

52

11

8

105

15

15

32

Pra

ctic

e (

Ave

rag

e)

Su

rfac

e A

reas

of

Pri

sms

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3



Readin

g t

o L

earn

Math

em

ati

csS

urf

ace

Are

as o

f P

rism

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3

©G

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eom

etry

Lesson 12-3

Pre-

Act

ivit

yH

ow d

o b

rick

mas

ons

kn

ow h

ow m

any

bri

cks

to o

rder

for

a p

roje

ct?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-3 a

t th

e to

p of

pag

e 64

9 in

you

r te

xtbo

ok.

How

cou

ld t

he

bric

k m

ason

fig

ure

ou

t ap

prox

imat

ely

how

man

y br

icks

are

nee

ded

for

the

gara

ge?

Sam

ple

an

swer

:F

ind

th

e ar

ea o

f ea

ch w

all

of

the

gar

age

and

ad

d t

hes

e ar

eas.

Div

ide

this

to

tal a

rea

by t

he

area

of

on

e fa

ce o

f a

bri

ck,b

ut

take

into

acc

ou

nt

the

amo

un

to

f sp

ace

for

the

mo

rtar

bet

wee

n t

he

bri

cks.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h s

ente

nce

is

alw

ays,

som

etim

es,o

r n

ever

tru

e.

a.A

bas

e of

a p

rism

is

a fa

ce o

f th

e pr

ism

.al

way

sb

.A

fac

e of

a p

rism

is

a ba

se o

f th

e pr

ism

.so

met

imes

c.T

he

late

ral

face

s of

a p

rism

are

rec

tan

gles

.so

met

imes

d.

If a

bas

e of

a p

rism

has

nve

rtic

es,t

hen

th

e pr

ism

has

nfa

ces.

nev

ere.

If a

bas

e of

a p

rism

has

nve

rtic

es,t

hen

th

e pr

ism

has

nla

tera

l ed

ges.

alw

ays

f.In

a r

igh

t pr

ism

,th

e la

tera

l ed

ges

are

also

alt

itu

des.

alw

ays

g.T

he

base

s of

a p

rism

are

con

gru

ent

regu

lar

poly

gon

s.so

met

imes

h.

An

y tw

o la

tera

l ed

ges

of a

pri

sm a

re p

erpe

ndi

cula

r to

eac

h o

ther

.n

ever

i.In

a r

ecta

ngu

lar

pris

m,a

ny

pair

of

oppo

site

fac

es c

an b

e ca

lled

th

e ba

ses.

alw

ays

j.A

ll o

f th

e la

tera

l fa

ces

of a

pri

sm a

re c

ongr

uen

t to

eac

h o

ther

.so

met

imes

2.E

xpla

in t

he

diff

eren

ce b

etw

een

th

e la

tera

l ar

eaof

a p

rism

an

d th

e su

rfac

e ar

eaof

apr

ism

.You

r ex

plan

atio

n s

hou

ld a

pply

to

both

rig

ht

and

obli

que

pris

ms.

Do

not

use

an

yfo

rmu

las

in y

our

expl

anat

ion

.S

amp

le a

nsw

er:T

he

late

ral a

rea

is t

he

sum

of

the

area

s o

f la

tera

l fac

es.T

he

surf

ace

area

is t

he

sum

of

the

area

s o

f al

lth

e fa

ces,

incl

ud

ing

bo

th t

he

late

ral f

aces

an

d t

he

bas

es.

3.R

efer

to

the

figu

re.

a.N

ame

this

sol

id w

ith

as

spec

ific

a n

ame

as p

ossi

ble.

rig

ht

pen

tag

on

al p

rism

b.

Nam

e th

e ba

ses

of t

he

soli

d.p

enta

go

ns

AB

CD

Ean

d F

GH

IJc.

Nam

e th

e la

tera

l fa

ces.

rect

ang

les

AB

GF

,BC

HG

,CD

IH,D

EJI

,an

d E

AF

Jd

.N

ame

the

edge

s.A�

F�,B�

G�,C�

H�,D�

I�,E�

J�,A�

B�,B�

C�,C�

D�,D�

E�,E�

A�,F�

G�,G�

H�,H�

I�,I�J�

,J�F�

e.N

ame

an a

ltit

ude

of

the

soli

d.A�

F�,B�

G�,C�

H�,D�

I�,o

r E�

J�f.

If a

repr

esen

ts t

he

area

,Pre

pres

ents

th

e pe

rim

eter

of

one

of t

he

base

s,an

d x

�A

F,

wri

te a

n ex

pres

sion

for

the

sur

face

are

a of

the

sol

id t

hat

invo

lves

a,P

,and

x.

Px

�2a

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is

to r

elat

e it

to

an e

very

day

use

of

the

sam

e w

ord.

How

can

th

e w

ay t

he

wor

d la

tera

lis

use

d in

spo

rts

hel

p yo

u r

emem

ber

the

mea

nin

g of

th

e la

tera

l ar

eaof

a s

olid

?S

amp

le a

nsw

er:

In f

oo

tbal

l,a

pas

sth

row

n t

o t

he

sid

eis

cal

led

a la

tera

l pas

s.In

geo

met

ry,t

he

late

ral a

rea

isth

e “s

ide

area

,”o

r th

e su

m o

f th

e ar

eas

of

all t

he

sid

e su

rfac

es o

f th

eso

lid (

the

surf

aces

oth

er t

han

th

e b

ases

).

A

BC

D IF

E

GH

J

©G

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eom

etry

Cro

ss S

ecti

on

s o

f P

rism

sW

hen

a p

lan

e in

ters

ects

a s

olid

fig

ure

to

form

a

two-

dim

ensi

onal

fig

ure

,th

e re

sult

s is

cal

led

a cr

oss

sect

ion

.Th

e fi

gure

at

the

righ

t sh

ows

a pl

ane

inte

rsec

tin

g a

cube

.Th

e cr

oss

sect

ion

is

a h

exag

on.

For

eac

h r

igh

t p

rism

,con

nec

t th

e la

bel

ed p

oin

ts i

n a

lph

abet

ical

or

der

to

show

a c

ross

sec

tion

.Th

en i

den

tify

th

e p

olyg

on.

1.2.

3.

rect

ang

letr

ian

gle

trap

ezo

id

Ref

er t

o th

e ri

ght

pri

sms

show

n a

t th

e ri

ght.

In t

he

rect

angu

lar

pri

sm,A

and

Car

e m

idp

oin

ts.I

den

tify

th

e cr

oss-

sect

ion

p

olyg

on f

orm

ed b

y a

pla

ne

con

tain

ing

the

give

n p

oin

ts.

4.A

,C,H

rect

ang

le

5.C

,E,G

tria

ng

le

6.H

,C,E

,Ftr

apez

oid

7.H

,A,E

pen

tag

on

8.B

,D,F

rect

ang

le

9.V

,X,R

tria

ng

le

10.R

,T,Y

rect

ang

le

11.R

,S,W

trap

ezo

id

A

B HG

F

DC

E

VXS

U

Y

Z

R

T

W

AB

CD

X

Y Z

S

R

U

T

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-3

12-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Su

rfac

e A

reas

of

Cyl

ind

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4

©G

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ill67

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eom

etry

Lesson 12-4

Late

ral A

reas

of

Cyl

ind

ers

A c

ylin

der

is a

sol

id w

hos

e ba

ses

are

con

gru

ent

circ

les

that

lie

in

par

alle

l pl

anes

.Th

e ax

isof

a c

ylin

der

is t

he

segm

ent

wh

ose

endp

oin

ts a

re t

he

cen

ters

of

thes

e ci

rcle

s.F

or a

rig

ht

cyli

nd

er,t

he

axis

an

d th

e al

titu

de o

f th

e cy

lin

der

are

equ

al.T

he

late

ral

area

of

a ri

ght

cyli

nde

r is

th

eci

rcu

mfe

ren

ce o

f th

e cy

lin

der

mu

ltip

lied

by

the

hei

ght.

Lat

eral

Are

a If

a cy

linde

r ha

s a

late

ral a

rea

of L

squa

re u

nits

, a

heig

ht o

f h

units

, o

f a

Cyl

ind

eran

d th

e ba

ses

have

rad

ii of

run

its,

then

L�

2�rh

.

Fin

d t

he

late

ral

area

of

the

cyli

nd

er a

bov

e if

th

e ra

diu

s of

th

e b

ase

is 6

cen

tim

eter

s an

d t

he

hei

ght

is 1

4 ce

nti

met

ers.

L�

2�rh

Late

ral a

rea

of a

cyl

inde

r

�2�

(6)(

14)

Sub

stitu

tion

�52

7.8

Sim

plify

.

Th

e la

tera

l ar

ea i

s ab

out

527.

8 sq

uar

e ce

nti

met

ers.

Fin

d t

he

late

ral

area

of

each

cyl

ind

er.R

oun

d t

o th

e n

eare

st t

enth

.

1.2.

301.

6 cm

237

7.0

in2

3.4.

113.

1 cm

250

2.7

cm2

5.6.

150.

8 m

212

.6 m

2

2 m

1 m

12 m

4 m

20 c

m

8 cm

6 cm

3 cm

3 cm

6 in

.10

in.

12 c

m

4 cm

radi

us o

f bas

eax

is

base

base

heig

ht

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill68

0G

lenc

oe G

eom

etry

Surf

ace

Are

as o

f C

ylin

der

sT

he

surf

ace

area

of

a cy

lin

der

is t

he

late

ral

area

of

the

cyli

nde

r pl

us

the

area

s of

th

e ba

ses.

Su

rfac

e A

rea

If a

cylin

der

has

a su

rfac

e ar

ea o

f T

squa

re u

nits

, a

heig

ht o

f o

f a

Cyl

ind

erh

units

, an

d th

e ba

ses

have

rad

ii of

run

its,

then

T�

2�rh

�2�

r2.

Fin

d t

he

surf

ace

area

of

the

cyli

nd

er.

Fin

d th

e la

tera

l ar

ea o

f th

e cy

lin

der.

If t

he

diam

eter

is

12 c

enti

met

ers,

then

th

e ra

diu

s is

6 c

enti

met

ers.

L�

Ph

Late

ral a

rea

of a

cyl

inde

r

�(2

�r)

hP

�2�

r

�2�

(6)(

14)

r�

6, h

�14

�52

7.8

Sim

plify

.

Fin

d th

e ar

ea o

f ea

ch b

ase.

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3.1

Sim

plify

.

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tal

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the

late

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area

plu

s th

e ar

ea o

f th

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o ba

ses.

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754

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cen

tim

eter

s.

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he

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ace

area

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cyl

ind

er.R

oun

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eare

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enth

.

1.2.

603.

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250

.3 m

2

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yd

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6 m

214

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.

8 in

.

2 m

15 m

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.

12 in

.

2 yd

3 yd

2 m

2 m

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.

10 in

.

14 c

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12 c

m

base

base

late

ral a

rea

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Su

rfac

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ind

ers

NA

ME

____

____

____

____

____

____

____

____

____

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____

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AT

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____

____

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ER

IOD

____

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12-4

12-4

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Su

rfac

e A

reas

of

Cyl

ind

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

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.

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he

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ace

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uar

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ers,

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ght

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he

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ace

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ght

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s.

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4 m

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.

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ght

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he

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.

3 in

.

12.K

ALE

IDO

SCO

PES

Nat

han

bu

ilt

a ka

leid

osco

pe w

ith

a 2

0-ce

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er b

arre

l an

d a

5-ce

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er d

iam

eter

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plan

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cov

er t

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ith

em

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aper

of

his

ow

nde

sign

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man

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aper

wil

l it

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e to

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el o

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30 m

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.

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.

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e (

Ave

rag

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Su

rfac

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reas

of

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NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

12-4

12-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csS

urf

ace

Are

as o

f C

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s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

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12-4

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Pre-

Act

ivit

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ow a

re c

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s u

sed

in

ext

rem

e sp

orts

?R

ead

the

intr

oduc

tion

to

Les

son

12-4

at

the

top

of p

age

655

in y

our

text

book

.

Wh

y is

th

e su

rfac

e ar

ea o

f a

hal

f-pi

pe m

ore

than

hal

f th

e su

rfac

e ar

ea o

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com

plet

e pi

pe?

Sam

ple

an

swer

:Th

e su

rfac

e ar

ea o

f th

e h

alf-

pip

e in

clu

des

an

add

ed f

lat

sect

ion

in t

he

mid

dle

.

Rea

din

g t

he

Less

on

1.U

nde

rlin

e th

e co

rrec

t w

ord

or p

hra

se t

o fo

rm a

tru

e st

atem

ent.

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he

base

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a c

ylin

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are

(rec

tan

gles

/reg

ula

r po

lygo

ns/

circ

les)

.

b.

Th

e (a

xis/

radi

us/

diam

eter

) of

a c

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he

segm

ent

wh

ose

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re t

he

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ters

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he

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ompo

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con

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d on

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mic

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e).

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In a

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ht

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lso

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ase/

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ral

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/alt

itu

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cyl

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s n

ot a

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cyl

inde

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atch

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lum

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ts v

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the

late

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a ri

ght

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nde

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wh

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th

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diu

s of

each

bas

e is

xcm

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d th

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ngt

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f th

e ax

is i

s y

cmv

b.

the

surf

ace

area

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a ri

ght

pris

m w

ith

squ

are

base

s in

wh

ich

the

len

gth

of

a si

de o

f a

base

is

xcm

an

d th

e le

ngt

h o

f a

late

ral

edge

is

ycm

ic.

the

surf

ace

area

of

a ri

ght

cyli

nde

r in

wh

ich

th

e ra

diu

s of

aba

se i

s x

cm a

nd

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hei

ght

is y

cmii

d.

the

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ace

area

of

regu

lar

hex

ahed

ron

(cu

be)

in w

hic

h t

he

len

gth

of

each

edg

e is

xcm

ive.

the

late

ral

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a tr

ian

gula

r pr

ism

in

wh

ich

th

e ba

ses

are

equ

ilat

eral

tri

angl

es w

ith

sid

e le

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h x

cm a

nd

the

hei

ght

isy

cmiii

f.th

e su

rfac

e ar

ea o

f a

righ

t cy

lin

der

in w

hic

h t

he

diam

eter

of

the

base

is

xcm

an

d th

e le

ngt

h o

f th

e ax

is i

s y

cmvi

i.(2

x2�

4xy)

cm

2

ii.(

2�xy

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cm2

iii.

3xy

cm2

iv.

6x2

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v.2�

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vi. ��

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�xy

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pin

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ou

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emb

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en t

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best

way

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rem

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mat

hem

atic

al f

orm

ula

is

to t

hin

k ab

out

wh

ere

the

diff

eren

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rts

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he f

orm

ula

com

e fr

om.H

ow c

an y

ou u

se t

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appr

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to

rem

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r th

efo

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a fo

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amp

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or

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two

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sum

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thei

r ar

eas

is 2

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.Th

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um

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nce

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f th

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t h

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the

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,so

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tera

l are

a is

2�

rh.A

dd

th

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tera

l are

a an

d t

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the

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et t

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area

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.

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ever

age

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ath

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idea

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ome

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al t

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um

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nce

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enti

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n b

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ach

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in

Exe

rcis

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xerc

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16.

Exe

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e 2

7.E

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38.

Exe

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ght

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you

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t a?

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.

10.F

ind

the

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nt

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sed

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En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-4

12-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Su

rfac

e A

reas

of

Pyr

amid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-5

12-5

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Late

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Reg

ula

r Py

ram

ids

Her

e ar

e so

me

prop

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es o

f py

ram

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he

base

is

a po

lygo

n.

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ll o

f th

e fa

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exce

pt t

he

base

,in

ters

ect

in a

com

mon

poi

nt

know

n a

s th

e ve

rtex

.•

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e fa

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that

in

ters

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at t

he

vert

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hic

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re c

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tera

l fa

ces,

are

tria

ngl

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egu

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e ba

se i

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an

d th

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ant

hei

ght

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�.

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the

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96

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.

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l ar

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e fe

et.

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he

late

ral

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r p

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ry.

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180

cm2

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5.6.

701.

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221

6 yd

2

45�

12 y

d60

�18

in.

60�

6 ft

42 m

20 m

3.5

ft

10 ft

8 cm

8 cm

8 cm

15 c

m

base

late

ral e

dge

late

ral f

ace

verte

x

slan

t hei

ght

Exer

cises

Exer

cises

Exam

ple

Exam

ple

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Surf

ace

Are

as o

f R

egu

lar

Pyra

mid

sT

he

surf

ace

area

of

a r

egu

lar

pyra

mid

is

the

late

ral

area

plu

s th

e ar

ea o

f th

e ba

se.

Su

rfac

e A

rea

of

aIf

a re

gula

r py

ram

id h

as a

sur

face

are

a of

Tsq

uare

uni

ts,

Reg

ula

r P

yram

ida

slan

t he

ight

of

�un

its,

and

its b

ase

has

a pe

rimet

er o

f P

units

and

an

area

of

Bsq

uare

uni

ts,

then

T�

�1 2� P�

�B

.

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th

e re

gula

r sq

uar

e p

yram

id a

bov

e,fi

nd

th

e su

rfac

e ar

ea t

o th

en

eare

st t

enth

if

each

sid

e of

th

e b

ase

is 1

2 ce

nti

met

ers

and

th

e h

eigh

t of

th

ep

yram

id i

s 8

cen

tim

eter

s.L

ook

at t

he

pyra

mid

abo

ve.T

he

slan

t h

eigh

t is

th

e h

ypot

enu

se o

f a

righ

t tr

ian

gle.

On

e le

g of

that

tri

angl

e is

th

e h

eigh

t of

th

e py

ram

id,a

nd

the

oth

er l

eg i

s h

alf

the

len

gth

of

a si

de o

f th

eba

se.U

se t

he

Pyt

hag

orea

n T

heo

rem

to

fin

d th

e sl

ant

hei

ght

�.

�2�

62�

82P

ytha

gore

an T

heor

em

�10

0S

impl

ify.

��

10Ta

ke t

he s

quar

e ro

ot o

f ea

ch s

ide.

T�

�1 2� P�

�B

Sur

face

are

a of

a p

yram

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4)(1

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0) �

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12),

��

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B�

122

�38

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impl

ify.

Th

e su

rfac

e ar

ea i

s 38

4 sq

uar

e ce

nti

met

ers.

Fin

d t

he

surf

ace

area

of

each

reg

ula

r p

yram

id.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

547.

4 cm

261

8.0

ft2

3.4.

400

cm2

456.

8 in

2

5.6.

360

cm2

340

yd2

12 y

d

10 y

d

12 c

m13

cm

6 in

.8.

7 in

.15

in.

60�

10 c

m

45�

8 ft

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m

20 c

m

late

ral e

dge

base

slan

t hei

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heig

ht

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Su

rfac

e A

reas

of

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s

NA

ME

____

____

____

____

____

____

____

____

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____

____

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ER

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____

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12-5

12-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Su

rfac

e A

rea

of

Pyr

amid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

____

_

12-5

12-5

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Lesson 12-5

Fin

d t

he

surf

ace

area

of

each

reg

ula

r p

yram

id.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

72 c

m2

646.

3 in

2

3.4.

455.

3 m

258

5.0

ft2

5.6.

99.9

mm

212

0 yd

2

7.8.

864

m2

945.

3 in

2

16 in

.

20 in

.

18 m

12 m

6 yd

7 yd

9 m

m

6 m

m

14 ft

12 ft

9 m 10

m

20 in

.

8 in

.4

cm

7 cm

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Fin

d t

he

surf

ace

area

of

each

reg

ula

r p

yram

id.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

261

yd2

147.

2 m

2

3.4.

205.

5 ft

276

.2 c

m2

5.6.

322.

2 in

278

4.7

mm

2

7.8.

75.7

yd

213

0.1

m2

9.G

AZE

BO

ST

he

roof

of

a ga

zebo

is

a re

gula

r oc

tago

nal

pyr

amid

.If

the

base

of

the

pyra

mid

has

sid

es o

f 0.

5 m

eter

s an

d th

e sl

ant

hei

ght

of t

he

roof

is

1.9

met

ers,

fin

d th

ear

ea o

f th

e ro

of.

3.8

m2

4.5

m

12 m

5.2

yd

3 yd

15 m

m

12 m

m12

in.

11 in

.

8 cm

2.5

cm

13 ft

5 ft

12 m

7 m

9 yd

10 y

d

Pra

ctic

e (

Ave

rag

e)

Su

rfac

e A

rea

of

Pyr

amid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

12-5

12-5



Readin

g t

o L

earn

Math

em

ati

csS

urf

ace

Are

as o

f P

yram

ids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-5

12-5

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Lesson 12-5

Pre-

Act

ivit

yH

ow a

re p

yram

ids

use

d i

n a

rch

itec

ture

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-5 a

t th

e to

p of

pag

e 66

0 in

you

r te

xtbo

ok.

Why

do

you

thin

k th

at t

he a

rchi

tect

for

the

new

ent

ranc

e to

the

Lou

vre

deci

ded

to u

se a

pyr

amid

rat

her

than

a r

ecta

ngul

ar p

rism

?

Sam

ple

answ

er:T

he p

yram

id,w

ith it

s sh

arp

poin

t at

the

ver

tex

and

its s

lopi

ng s

ides

,is

mor

e un

usua

l and

dra

mat

ic t

han

are

ctan

gula

r pr

ism

.

Rea

din

g t

he

Less

on

1.In

th

e fi

gure

,AB

CD

Eh

as c

ongr

uen

t si

des

and

con

gru

ent

angl

es.

a.D

escr

ibe

this

pyr

amid

wit

h a

s sp

ecif

ic a

nam

e as

pos

sibl

e.re

gu

lar

pen

tag

on

al p

yram

id

b.

Use

th

e fi

gure

to

nam

e th

e ba

se o

f th

is p

yram

id.

AB

CD

E

c.D

escr

ibe

the

base

of

the

pyra

mid

.re

gu

lar

pen

tag

on

d.

Nam

e th

e ve

rtex

of

the

pyra

mid

.P

e.N

ame

the

late

ral f

aces

of

the

pyra

mid

.�

PAB

,�P

BC

,�P

CD

,�P

DE

,an

d �

PE

A

f.D

escr

ibe

the

late

ral

face

s.fi

ve c

on

gru

ent

iso

scel

es t

rian

gle

s

g.N

ame

the

late

ral

edge

s of

th

e py

ram

id.

P�A�

,P�B�

,P�C�

,P�D�

,an

d P�

E�h

.N

ame

the

alti

tude

of

the

pyra

mid

.P�

Q�i.

Wri

te a

n e

xpre

ssio

n f

or t

he

hei

ght

of t

he

pyra

mid

.P

Q

j.W

rite

an

exp

ress

ion

for

th

e sl

ant

hei

ght

of t

he

pyra

mid

.P

S

2.In

a r

egu

lar

squ

are

pyra

mid

,let

sre

pres

ent

the

side

len

gth

of

the

base

,hre

pres

ent

the

hei

ght,

are

pres

ent

the

apot

hem

,an

d �

repr

esen

t th

e sl

ant

hei

ght.

Als

o,le

t L

repr

esen

tth

e la

tera

l ar

ea a

nd

let

Tre

pres

ent

the

surf

ace

area

.Wh

ich

of

the

foll

owin

gre

lati

onsh

ips

are

corr

ect?

A,D

,E,F

A.

s�

2aB

.a2

��2

�h

2C

.L

�4�

s

D.h

��

�2�

a�

2 �E

.�� 2s � �2

�h

2�

�2F.

T�

s2�

2�s

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

eth

ing

is t

o ex

plai

n i

t to

som

eon

e el

se.S

upp

ose

that

on

e of

you

r cl

assm

ates

is

hav

ing

trou

ble

rem

embe

rin

g th

e di

ffer

ence

bet

wee

n t

he

hei

ght

and

the

slan

t h

eigh

tof

a r

egu

lar

pyra

mid

.How

can

you

exp

lain

th

is c

once

pt?

Sam

ple

an

swer

:Th

e he

ight

of

the

pyra

mid

is t

he

len

gth

of

the

per

pen

dic

ula

r se

gm

ent

fro

m t

he

vert

ex t

o t

he

cen

ter

of

the

bas

e.T

he

slan

the

ight

is t

he

len

gth

of

the

per

pen

dic

ula

r se

gm

ent

fro

m t

he

vert

ex o

f th

epy

ram

id t

o t

he

mid

po

int

of

on

e o

f th

e si

des

of

the

bas

e o

f th

e py

ram

id.

ED

CA S

B

QP

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Two

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To

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te a

tru

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d,yo

u c

ould

sta

rt w

ith

an

ord

inar

y so

lid

and

then

cu

t of

f th

eco

rner

s.A

not

her

way

to

mak

e su

ch a

sh

ape

is t

o u

se t

he

patt

ern

s on

th

is p

age.

Th

e T

run

cate

d O

ctah

edro

n

1.T

wo

copi

es o

f th

e pa

tter

n a

t th

e ri

ght

can

be

use

d to

mak

e a

tru

nca

ted

oct

ahed

ron

,a s

olid

w

ith

6 s

quar

e fa

ces

and

8 re

gula

r h

exag

onal

fa

ces.

Eac

h p

atte

rn m

akes

hal

f of

th

e tr

un

cate

doc

tah

edro

n.A

ttac

h a

djac

ent

face

s u

sin

g gl

ue

or t

ape

to m

ake

a cu

p-sh

aped

fig

ure

.

Th

e T

run

cate

d T

etra

hed

ron

2.T

he

patt

ern

bel

ow w

ill

mak

e a

tru

nca

ted

tet

rah

edro

n,

a so

lid

wit

h 8

pol

ygon

al f

aces

:4 h

exag

ons

and

4 eq

uil

ater

al t

rian

gles

.

Sol

ve.

3.F

ind

the

surf

ace

area

of

the

tru

nca

ted

octa

hed

ron

if

each

pol

ygon

in

th

e pa

tter

n h

as

side

s of

3 i

nch

es.

241.

1 in

2

4.F

ind

the

surf

ace

area

of

the

tru

nca

ted

tetr

ahed

ron

if

each

pol

ygon

in

th

e pa

tter

n h

as

side

s of

3 i

nch

es.

109.

1 in

2

Are

a F

orm

ula

s fo

r R

egu

lar

Po

lyg

on

s(s

is t

he le

ngth

of

one

side

)

tria

ngle

A�

�s 42 ��

3�

hexa

gon

A�

�3 2s2 ��

3�

octa

gon

A�

2s2 ( �

2��

1)

Tape

or

glue

her

e.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-5

12-5



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Su

rfac

e A

reas

of

Co

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-6

12-6

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Lesson 12-6

Late

ral A

reas

of

Co

nes

Con

es h

ave

the

foll

owin

g pr

oper

ties

.•

A c

one

has

on

e ci

rcu

lar

base

an

d on

e ve

rtex

.•

Th

e se

gmen

t w

hos

e en

dpoi

nts

are

th

e ve

rtex

an

d th

e ce

nte

r of

th

e ba

se i

s th

e ax

isof

th

e co

ne.

•T

he

segm

ent

that

has

on

e en

dpoi

nt

at t

he

vert

ex,i

s pe

rpen

dicu

lar

to t

he

base

,an

d h

as i

ts o

ther

en

dpoi

nt

on t

he

base

is

the

alti

tud

eof

th

e co

ne.

•F

or a

rig

ht

con

eth

e ax

is i

s al

so t

he

alti

tude

,an

d an

y se

gmen

t fr

om t

he

circ

um

fere

nce

of

th

e ba

se t

o th

e ve

rtex

is

the

slan

t h

eigh

t�.

If a

con

e is

not

a r

igh

t co

ne,

it i

s ob

liqu

e.

Lat

eral

Are

a If

a co

ne h

as a

late

ral a

rea

of L

squa

re u

nits

, a

slan

t he

ight

of

�un

its,

of

a C

on

ean

d th

e ra

dius

of

the

base

is r

units

, th

en L

��

r�.

Fin

d t

he

late

ral

area

of

a co

ne

wit

h s

lan

t h

eigh

t of

10

cen

tim

eter

s an

d a

bas

e w

ith

a r

adiu

s of

6 c

enti

met

ers.

L�

�r �

Late

ral a

rea

of a

con

e

��

(6)(

10)

r�

6, �

�10

�18

8.5

Sim

plify

.

Th

e la

tera

l ar

ea i

s ab

out

188.

5 sq

uar

e ce

nti

met

ers.

Fin

d l

ater

al a

rea

of e

ach

cir

cula

r co

ne.

Rou

nd

to

the

nea

rest

ten

th.

1.2.

47.1

cm

242

4.1

cm2

3.4.

391.

8 m

281

6.8

mm

2

5.6.

204.

2 in

228

4.3

yd2

8 yd

16 y

d12

in.

5 in

.

26 m

m

20 m

m

6��3

m

60�

9 cm

15 c

m

3 cm

4 cm

6 cm

10 c

m

axis

base

base

�sl

ant h

eigh

t

right

con

eob

lique

con

e

altit

ude

VV

Exer

cises

Exer

cises

Exam

ple

Exam

ple

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etry

Surf

ace

Are

as o

f C

on

esT

he

surf

ace

area

of

a co

ne

is t

he

late

ral

area

of

the

con

e pl

us

the

area

of

the

base

.

Su

rfac

e A

rea

If a

cone

has

a s

urfa

ce a

rea

of T

squa

re u

nits

, a

slan

t he

ight

of

of

a R

igh

t C

on

e�

units

, an

d th

e ra

dius

of

the

base

is r

units

, th

en T

��

r��

�r2

.

For

th

e co

ne

abov

e,fi

nd

th

e su

rfac

e ar

ea t

o th

e n

eare

st t

enth

if

the

rad

ius

is 6

cen

tim

eter

s an

d t

he

hei

ght

is 8

cen

tim

eter

s.T

he

slan

t h

eigh

t is

th

e h

ypot

enu

se o

f a

righ

t tr

ian

gle

wit

h l

egs

of l

engt

h 6

an

d 8.

Use

th

eP

yth

agor

ean

Th

eore

m.

�2�

62�

82P

ytha

gore

an T

heor

em

�2�

100

Sim

plify

.

��

10Ta

ke t

he s

quar

e ro

ot o

f ea

ch s

ide.

T�

�r�

��

r2S

urfa

ce a

rea

of a

con

e

��

(6)(

10)

��

�62

r�

6, �

�10

�30

1.6

Sim

plify

.

Th

e su

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out

301.

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e ce

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d t

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e.R

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enth

.

1.2.

678.

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m

40 m

4 in

.

45�

12 c

m

13 c

m

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9 cm

12 c

m

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t

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r

Stu

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uid

e a

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(con

tinued

)

Su

rfac

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of

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nes

NA

ME

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____

____

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____

____

____

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12-6

12-6

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

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Su

rfac

e A

reas

of

Co

nes

NA

ME

____

____

____

____

____

____

____

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12-6

12-6

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Lesson 12-6

Fin

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of

each

con

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if

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ry.

1.2.

298.

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60.1

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3.4.

728.

8 in

273

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2

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661.

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10.F

ind

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264

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ind

the

surf

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if t

he

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ght

is 1

4 ce

nti

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and

the

slan

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16.4

cen

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s.

669.

3 cm

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9.F

ind

the

surf

ace

area

of

a co

ne

if t

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ght

is 1

2 in

ches

an

d th

e di

amet

er i

s 27

in

ches

.

1338

.6 in

2

10.H

ATS

Cuo

ng b

ough

t a

coni

cal

hat

on a

rec

ent

trip

to

cent

ral V

ietn

am.T

he b

asic

fra

me

ofth

e h

at i

s 16

hoo

ps o

f ba

mbo

o th

at g

radu

ally

dim

inis

h i

n s

ize.

Th

e h

at i

s co

vere

d in

pal

mle

aves

.If

the

hat

has

a d

iam

eter

of

50 c

enti

met

ers

and

a sl

ant

hei

ght

of 3

2 ce

nti

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wh

at i

s th

e la

tera

l ar

ea o

f th

e co

nic

al h

at?

abo

ut

2513

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7 cm

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m4

m

3 ft

19 ft

48 y

d25 y

d

7 m

m18

mm

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.13

in.

Pra

ctic

e (

Ave

rag

e)

Su

rfac

e A

reas

of

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nes

NA

ME

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____

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____

____

____

____

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12-6

12-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csS

urf

ace

Are

as o

f C

on

es

NA

ME

____

____

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____

____

____

____

____

____

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12-6

12-6

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Lesson 12-6

Pre-

Act

ivit

yH

ow i

s th

e la

tera

l ar

ea o

f a

con

e u

sed

to

cove

r te

pee

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 12

-6 a

t th

e to

p of

pag

e 66

6 in

you

r te

xtbo

ok.

If y

ou w

ante

d to

bu

ild

a te

pee

of a

cer

tain

siz

e,h

ow w

ould

it

hel

p yo

u t

okn

ow t

he

form

ula

for

th

e la

tera

l ar

ea o

f a

con

e?S

amp

le a

nsw

er:T

he

form

ula

wo

uld

hel

p y

ou

est

imat

e h

ow

man

y an

imal

ski

ns

you

wo

uld

nee

d f

or

the

tep

ee.

Rea

din

g t

he

Less

on

1.

a.W

hic

h n

et w

ill

give

th

e co

ne

wit

h t

he

grea

test

lat

eral

are

a?N

et B

b.

Wh

ich

net

wil

l gi

ve t

he

tall

est

con

e?N

et C

2.R

efer

to

the

figu

re a

t th

e ri

ght.

Su

ppos

e yo

u h

ave

rem

oved

th

e ci

rcu

lar

base

of

the

con

e an

d cu

t fr

om V

to A

so t

hat

you

can

un

roll

th

e la

tera

lsu

rfac

e on

to a

fla

t ta

ble.

a.H

ow c

an y

ou b

e su

re t

hat

th

e fl

atte

ned

-ou

t pi

ece

is a

sec

tor

of a

cir

cle?

Sam

ple

an

swer

:B

efo

re y

ou

un

roll

the

late

ral s

urf

ace,

all

po

ints

on

th

e b

ott

om

rim

of

the

con

e ar

e �

un

its

fro

m V

.A

fter

yo

u f

latt

en o

ut

the

surf

ace,

tho

se p

oin

ts a

re s

till

�u

nit

s fr

om

V.T

his

mea

ns

that

th

ey w

ill li

e o

n a

cir

cle

wit

h

cen

ter

Van

d r

adiu

s �.

b.

How

do

you

kn

ow t

hat

th

e fl

atte

ned

-ou

t pi

ece

is n

ot a

fu

ll c

ircl

e?S

amp

le a

nsw

er:

r�

�

3.S

upp

ose

you

hav

e a

righ

t co

ne

wit

h r

adiu

s r,

diam

eter

d,h

eigh

t h

,an

d sl

ant

hei

ght

�.W

hic

h o

f th

e fo

llow

ing

rela

tion

ship

s in

volv

ing

thes

e le

ngt

hs

are

corr

ect?

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A.

r�

2dB

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h�

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ou

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emb

er

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ne

way

to

rem

embe

r a

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for

mu

la i

s to

rel

ate

it t

o a

form

ula

you

alr

eady

kn

ow.

Exp

lain

how

th

e fo

rmu

las

for

the

late

ral

area

s of

a p

yram

id a

nd

a co

ne

are

sim

ilar

.S

amp

le a

nsw

er:

Bo

th f

orm

ula

s sa

y th

at t

he

late

ral a

rea

incl

ud

es t

he

dis

tan

ce a

rou

nd

th

e b

ase

mu

ltip

lied

by

the

slan

t h

eig

ht.

V

PA

B

h r

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Net

AN

et B

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C

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Th

e p

atte

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t th

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ade

from

a

circ

le.I

t ca

n b

e fo

lded

to

mak

e a

con

e.

1.M

easu

re t

he

radi

us

of t

he

circ

le t

o th

e n

eare

st c

enti

met

er. 4

cm

2.T

he

patt

ern

is

wh

at f

ract

ion

of

the

com

plet

e ci

rcle

? �3 4�

3.W

hat

is

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circ

um

fere

nce

of

the

com

plet

e ci

rcle

? 8�

cm

4.H

ow l

ong

is t

he

circ

ula

r ar

c th

at i

s th

e ou

tsid

e of

th

e pa

tter

n?

6�cm

5.C

ut

out

the

patt

ern

an

d ta

pe i

t to

geth

er t

o fo

rm a

con

e.S

ee s

tud

ents

’wo

rk.

6.M

easu

re t

he

diam

eter

of

the

circ

ula

r ba

se o

f th

e co

ne.

6 cm

7.W

hat

is

the

circ

um

fere

nce

of

the

base

of

the

con

e? 6

�cm

8.W

hat

is

the

slan

t h

eigh

t of

th

e co

ne?

4 c

m

9.U

se t

he

Pyt

hag

orea

n T

heo

rem

to

calc

ula

te t

he

hei

ght

of t

he

con

e.U

se a

dec

imal

app

roxi

mat

ion

.Ch

eck

you

r ca

lcu

lati

on b

y m

easu

rin

g th

e h

eigh

t w

ith

a m

etri

c ru

ler.

2.65

cm

10.F

ind

the

late

ral

area

.12�

cm2

11.F

ind

the

tota

l su

rfac

e ar

ea.2

1�cm

2

Mak

e a

pap

er p

atte

rn f

or e

ach

con

e w

ith

th

e gi

ven

mea

sure

men

ts.

Th

en c

ut

the

pat

tern

ou

t an

d m

ake

the

con

e.F

ind

th

e m

easu

rem

ents

.

12.

13.

diam

eter

of

base

�8

cmdi

amet

er o

f ba

se �

10 c

m

late

ral

area

�24

�cm

2la

tera

l ar

ea �

50�

cm2

hei

ght

of c

one

�4.

5 cm

hei

ght

of c

one

�8.

7 cm

(to

nea

rest

ten

th o

f a

cen

tim

eter

)(t

o n

eare

st t

enth

of

a ce

nti

met

er)

20 c

m12

0�6

cm

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

12-6

12-6



Stu

dy G

uid

e a

nd I

nte

rven

tion

Su

rfac

e A

reas

of

Sp

her

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

12-7

12-7

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Lesson 12-7

Pro

per

ties

of

Sph

eres

A s

ph

ere

is t

he

locu

s of

all

poi

nts

th

at

are

a gi

ven

dis

tan

ce f

rom

a g

iven

poi

nt

call

ed i

ts c

ente

r.

Her

e ar

e so

me

term

s as

soci

ated

wit

h a

sph

ere.

•A

rad

ius

is a

seg

men

t w

hos

e en

dpoi

nts

are

th

e ce

nte

r of

th

e sp

her

e an

d a

poin

t on

th

e sp

her

e.•

A c

hor

dis

a s

egm

ent

wh

ose

endp

oin

ts a

re p

oin

ts

on t

he

sph

ere.

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dia

met

eris

a c

hor

d th

at c

onta

ins

the

sph

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s ce

nte

r.•

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ange

nt

is a

lin

e th

at i

nte

rsec

ts t

he

sph

ere

in

exac

tly

one

poin

t.•

A g

reat

cir

cle

is t

he

inte

rsec

tion

of

a sp

her

e an

d a

plan

e th

at c

onta

ins

the

cen

ter

of t

he

sph

ere.

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hem

isp

her

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on

e-h

alf

of a

sph

ere.

Eac

h g

reat

ci

rcle

of

a sp

her

e de

term

ines

tw

o h

emis

pher

es.

Det

erm

ine

the

shap

es y

ou g

et w

hen

you

in

ters

ect

a p

lan

e w

ith

a s

ph

ere.

The

inte

rsec

tion

of p

lane

MT

he in

ters

ectio

n of

pla

ne N

The

inte

rsec

tion

of p

lane

Pan

d sp

here

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poi

nt P

.an

d sp

here

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circ

le Q

.an

d sp

here

Ois

circ

le O

.

A p

lan

e ca

n i

nte

rsec

t a

sph

ere

in a

poi

nt,

in a

cir

cle,

or i

n a

gre

at c

ircl

e.

Des

crib

e ea

ch o

bje

ct a

s a

mod

el o

f a

circ

le,a

sp

her

e,a

hem

isp

her

e,or

non

e of

th

ese.

1.a

base

ball

2.a

pan

cake

3.th

e E

arth

sph

ere

circ

lesp

her

e

4.a

kett

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rill

cov

er5.

a ba

sket

ball

rim

6.co

la c

an

hem

isp

her

eci

rcle

no

ne

of

thes

e

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is t

rue

or f

als

e.

7.A

ll l

ines

in

ters

ecti

ng

a sp

her

e ar

e ta

nge

nt

to t

he

sph

ere.

fals

e

8.E

very

pla

ne

that

in

ters

ects

a s

pher

e m

akes

a g

reat

cir

cle.

fals

e

9.T

he

east

ern

hem

isph

ere

of E

arth

is

con

gru

ent

to t

he

wes

tern

hem

isph

ere.

tru

e

10.T

he

diam

eter

of

a sp

her

e is

con

gru

ent

to t

he

diam

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eat

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le.

tru

e

OP

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nt.

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circ

le th

at c

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ts S

, M, T

, and

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a gr

eat c

ircle

; it d

eter

min

es tw

o he

mis

pher

es.

chor

dta

ngen

t

grea

t circ

le

diam

eter

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usS

T

W

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V

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re

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ple

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Exer

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Surf

ace

Are

as o

f Sp

her

esYo

u c

an t

hin

k of

th

e su

rfac

e ar

ea o

f a

sph

ere

as t

he

tota

l ar

ea o

f al

l of

th

e n

onov

erla

ppin

g st

rips

it

wou

ld t

ake

to c

over

th

esp

her

e.If

ris

th

e ra

diu

s of

th

e sp

her

e,th

en t

he

area

of

a gr

eat

circ

le o

f th

esp

her

e is

�r2

.Th

e to

tal

surf

ace

area

of

the

sph

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is f

our

tim

es t

he

area

of

agr

eat

circ

le.

Su

rfac

e A

rea

of

a S

ph

ere

If a

sphe

re h

as a

sur

face

are

a of

Tsq

uare

uni

ts a

nd a

rad

ius

of r

units

, th

en T

�4�

r2.

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d t

he

surf

ace

area

of

a sp

her

e to

th

e n

eare

st t

enth

if

th

e ra

diu

s of

th

e sp

her

e is

6 c

enti

met

ers.

T�

4�r2

Sur

face

are

a of

a s

pher

e

�4�

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r�

6

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2.4

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plify

.

Th

e su

rfac

e ar

ea i

s 45

2.4

squ

are

cen

tim

eter

s.

Fin

d t

he

surf

ace

area

of

each

sp

her

e w

ith

th

e gi

ven

rad

ius

or d

iam

eter

to

the

nea

rest

ten

th.

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r�

2 �2�

ft

804.

2 cm

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d�

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enti

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ind

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pher

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ith

dia

met

er �

cen

tim

eter

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2

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ind

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if t

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(con

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)

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12-7

12-7

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Su

rfac

e A

reas

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Sp

her

es

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ME

____

____

____

____

____

____

____

____

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12-7

12-7

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Lesson 12-7

In t

he

figu

re,A

is t

he

cen

ter

of t

he

sph

ere,

and

pla

ne

Tin

ters

ects

th

e sp

her

e in

cir

cle

E.R

oun

d t

o th

e n

eare

stte

nth

if

nec

essa

ry.

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AE

�5

and

DE

�12

,fin

d A

D.

13

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AE

�7

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DE

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,fin

d A

D.

16.6

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th

e ra

diu

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th

e sp

her

e is

18

un

its

and

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radi

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of �

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17

un

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fin

d A

E.

5.9

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th

e ra

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s of

th

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her

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10

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its

and

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radi

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9 u

nit

s,fi

nd

AE

.

4.4

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t on

�E

and

AD

�23

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d A

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d t

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sp

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oun

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232

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rad

ius

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ards

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es

114.

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In t

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ters

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th

e sp

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cir

cle

R.R

oun

d t

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if

nec

essa

ry.

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CR

�4

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�14

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d C

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14.6

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CR

�7

and

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�24

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d C

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25

3.If

th

e ra

diu

s of

th

e sp

her

e is

28

un

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and

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radi

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a p

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and

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3,fi

nd

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sta

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ize

5 so

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es 1

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a t

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12-7

12-7



Readin

g t

o L

earn

Math

em

ati

csS

urf

ace

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as o

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12-7

12-7

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Lesson 12-7

Pre-

Act

ivit

yH

ow d

o m

anu

fact

ure

rs o

f sp

orts

eq

uip

men

t u

se t

he

surf

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area

s of

sp

her

es?

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d th

e in

trod

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on t

o L

esso

n 12

-7 a

t th

e to

p of

pag

e 67

1 in

you

r te

xtbo

ok.

How

wou

ld k

now

ing

the

form

ula

for

th

e su

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ea o

f a

sph

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hel

pm

anu

fact

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rs o

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ach

bal

ls?

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ple

an

swer

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e su

rfac

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ea o

fth

e b

all t

ells

th

e m

anu

fact

ure

r h

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mu

ch m

ater

ial i

s n

eed

ed t

om

ake

each

bal

l.

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din

g t

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Less

on

1.In

th

e fi

gure

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th

e ce

nte

r of

th

e sp

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e.N

ame

each

of

the

foll

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th

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ree

radi

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f.th

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2.D

eter

min

e w

het

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som

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nev

ertr

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pher

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d a

plan

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ters

ect

in m

ore

than

on

e po

int,

thei

r in

ters

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on w

ill

be a

grea

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rcle

.so

met

imes

b.

A g

reat

cir

cle

has

th

e sa

me

cen

ter

as t

he

sph

ere.

alw

ays

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he

endp

oin

ts o

f a

radi

us

of a

sph

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are

two

poin

ts o

n t

he

sph

ere.

nev

erd

.A

ch

ord

of a

sph

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is a

dia

met

er o

f th

e sp

her

e.so

met

imes

e.A

rad

ius

of a

gre

at c

ircl

e is

als

o a

radi

us

of t

he

sph

ere.

alw

ays

3.M

atch

eac

h s

urf

ace

area

for

mu

la w

ith

th

e n

ame

of t

he

appr

opri

ate

soli

d.

a.T

��

r��

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vii.

regu

lar

pyra

mid

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one

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emb

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Man

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hav

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oubl

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mem

beri

ng

all

of t

he

form

ula

s th

ey h

ave

lear

ned

in

th

isch

apte

r.W

hat

is

an e

asy

way

to

rem

embe

r th

e fo

rmu

la f

or t

he

surf

ace

area

of

a sp

her

e?S

amp

le a

nsw

er:

A s

ph

ere

do

esn

’t h

ave

any

late

ral f

aces

or

bas

es,s

o t

he

exp

ress

ion

in t

he

form

ula

fo

r it

s su

rfac

e ar

ea h

as ju

st o

ne

term

,4�

r2,

rath

er t

han

bei

ng

th

e su

m o

f th

e ex

pre

ssio

ns

for

the

late

ral a

rea

and

th

ear

ea o

f th

e b

ases

like

th

e o

ther

s.

RS

T

P

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Do

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ing

Siz

esC

onsi

der

wh

at h

appe

ns

to s

urf

ace

area

wh

en t

he

side

s of

a f

igu

re a

re d

oubl

ed.

Th

e si

des

of t

he

larg

e cu

be a

re t

wic

e th

e si

ze o

f th

e si

des

of t

he

smal

l cu

be.

1.H

ow l

ong

are

the

edge

s of

th

e la

rge

cube

?6

in.

2.W

hat

is

the

surf

ace

area

of

the

smal

l cu

be?

54 in

2

3.W

hat

is

the

surf

ace

area

of

the

larg

e cu

be?

216

in2

4.T

he

surf

ace

area

of

the

larg

e cu

be i

s h

ow m

any

tim

es g

reat

er

than

th

at o

f th

e sm

all

cube

?4

tim

es

Th

e ra

diu

s of

th

e la

rge

sph

ere

at t

he

righ

t is

tw

ice

the

radi

us

of

the

smal

l sp

her

e.

5.W

hat

is

the

surf

ace

area

of

the

smal

l sp

her

e?40

0� m

2

6.W

hat

is

the

surf

ace

area

of

the

larg

e sp

her

e? 1

600�

m2

7.T

he

surf

ace

area

of

the

larg

e sp

her

e is

how

man

y ti

mes

gr

eate

r th

an t

he

surf

ace

area

of

the

smal

l sp

her

e?4

tim

es

8.It

app

ears

th

at i

f th

e di

men

sion

s of

a s

olid

are

dou

bled

,th

e su

rfac

e ar

ea i

s m

ult

ipli

ed b

y ��

��

��.

4

Now

con

side

r h

ow d

oubl

ing

the

dim

ensi

ons

affe

cts

the

volu

me

of a

cu

be.

Th

e si

des

of t

he

larg

e cu

be a

re t

wic

e th

e si

ze o

f th

e sm

all

cube

.

9.H

ow l

ong

are

the

edge

s of

th

e la

rge

cube

?10

in.

10.W

hat

is

the

volu

me

of t

he

smal

l cu

be?

125

in3

11.W

hat

is

the

volu

me

of t

he

larg

e cu

be?

1000

in3

12.T

he

volu

me

of t

he

larg

e cu

be i

s h

ow m

any

tim

es g

reat

er t

han

th

at o

f th

e sm

all

cube

?8

tim

es

Th

e la

rge

sph

ere

at t

he

righ

t h

as t

wic

e th

e ra

diu

s of

th

e sm

all

sph

ere.

13.W

hat

is

the

volu

me

of t

he

smal

l sp

her

e?36

� m

3

14.W

hat

is

the

volu

me

of t

he

larg

e sp

her

e?28

8� m

3

15.T

he

volu

me

of t

he

larg

e sp

her

e is

how

man

y ti

mes

gre

ater

th

an t

he

volu

me

of t

he

smal

l sp

her

e?8

tim

es

16.I

t ap

pear

s th

at i

f th

e di

men

sion

s of

a s

olid

are

dou

bled

,th

e vo

lum

e is

mu

ltip

lied

by

��

��

�.8

3 m

5 in

.

3 m

5 in

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

12-7

12-7



Chapter 12 Assessment Answer Key Form 1 Form 2APage 703 Page 704 Page 705

(continued on the next page)

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

C

A

B

D

B

D

B

D

C

D

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

A

B

A

B

C

C

B

D

A

1020 ft2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

D

C

D

C

A

C

B

B

C

D


Chapter 12 Assessment Answer KeyForm 2A (continued) Form 2BPage 706 Page 707 Page 708

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

A

C

D

A

B

B

C

B

A

A

391.6 ft2

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

B

D

A

B

D

C

B

B

D

B

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

A

C

C

D

A

B

B

D

C

53 ft2


Chapter 12 Assessment Answer KeyForm 2CPage 709 Page 710

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

�PQRS, �PQT,�QTR, �RTS, �PTS

hexagonal pyramid

12

Sample answer:

240 in2

120 cm2

3536 units2

7.2 in2

7 gal

6 6

5 4 44

4

5

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

524.1 yd2

6 in.

270 in2

270 � 150�3� in2

144 units2

54.5 cm

62.8 ft2

113.1 ft2

17 m

235.6 cm2

38.5 in2


Chapter 12 Assessment Answer KeyForm 2DPage 711 Page 712

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

G�J�, H�J�, I�J�, G�H�,

H�I�, G�I�

pentagonalprism

12

Sample answer:

840 cm2

750 in2

258 ft2

3 gal

180 in2

88

8 8

88

6

66

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

845.2 m2

7 ft

540 ft2

931.0 ft2

156.9 units2

22.4 ft

204.2 cm2

282.7 cm2

5 m

1140.4 in2

113.1 in2


Chapter 12 Assessment Answer KeyForm 3Page 713 Page 714

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

�XYZ, �UVW

octahedron

20

Sample answer:

128 � 4�21�units2

35 � 7�13� ft2

528 units2

17,837.8 ft2

32,435.2 ft2

It is multiplied by 4.

52

2

2

22

2

2

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

468 in2

842.1 in2


233.8 in2

387.7 in2

182.0 cm2

2�21�

3�r2

1709 ft2

700 ft2


Chapter 12 Assessment Answer KeyPage 715, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofpyramids, prisms, cylinders, cones, spheres, surface area,lateral area, nets, and properties of solid figures.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures and drawings are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of pyramids,prisms, cylinders, cones, spheres, surface area, lateralarea, nets, and properties of solid figures.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures and drawings are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofpyramids, prisms, cylinders, cones, spheres, surface area,lateral area, nets, and properties of solid figures.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Figures and drawings are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work shown to substantiate the

final computation.• Figures and drawings may be accurate but lack detail or

explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the conceptsof pyramids, prisms, cylinders, cones, spheres, surfacearea, lateral area, nets, and properties of solid figures.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures and drawings are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations


An

swer

s

Chapter 12 Assessment Answer KeyPage 715, Open-Ended Assessment

Sample Answers

1. a.

b. e � f � v � 2

2. The lateral area is the area of the lateral faces. The surface area includesthe area of the lateral faces plus the areas of the two bases.

3. a.

b. �ABCD

c. �ABE, �BCE, �CDE, �ADE

d.

4.

5. a.

b. 38� in2

6. Sample answer: Sam is painting the walls of a room. The room is 12 feetlong, 10 feet wide, and 8 feet high. A gallon of paint covers 400 square feetand costs $16 per gallon. Find the cost to paint the room.

2 in. 6 in.3 in.

Oblique Right

A D

CB

E

Figure No. of Edges (e) No. of Faces (f ) No. of Vertices (v ) f � v

Triangular Pyramid 6 4 4 8

Triangular Prism 9 5 6 11

Cube 12 6 8 14

Square Pyramid 8 5 5 10

Hexagonal Prism 18 8 12 20

Hexagonal Pyramid 12 7 7 14

In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.


Chapter 12 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 716 Page 717 Page 718

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

slant height

cone

sphere

perspective view

Platonic Solids

prism

pyramid

surface area

right cylinder

hemisphere

the sum of the areasof the lateral faces

a prism whoselateral edges are

also altitudes

the intersection of asphere and a plane

containing thecenter of the sphere

Quiz 2Page 717

Quiz 4Page 718

1.

2.

3.

4.

Sample answer:

Sample answer:

D

1022

22

2

22 22

2

2222

2

1.

2.

3.

4.

5.

2 in.

12 units

rectangle

3870.4 units2

4630.7 units2

1.

2.

3.

4.

5.

81 units2

116.1 units2

260 cm2

188.5 ft2

267.0 ft2

1.

2.

3.

4.

5.

B�C�

AH��

�D

4071.5 in2



Chapter 12 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 719 Page 720

An

swer

s

Part I

Part II

6.

7.

8.

9.

10.

Sample answer:

1656 units2

92.8 units2

282.7 in2

6

5

5

4

4

4 53

1.

2.

3.

4.

5.

C

A

D

A

D

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

�AFD and �BFC

false

It alternates

between �12

� and 2.

m�X � 63,m�Z � 44,y � 34 cm

27.6 in. and 23.1 in.

magnitude: 11.7,direction: 149.0°

circle

28

80 cm2; 48 cm

14.5 m

5.0 in.


Chapter 12 Assessment Answer KeyStandardized Test Practice

Page 721 Page 722

1.

2.

3.

4.

5.

6.

7.

8. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D 9. 10.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

11.

12.

13.

14.

15.

16.

2.6 m

25

51.4 cm2

pentagonalpyramid

3141.6 in2

2206.2 cm2

1 4 5

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Chapter 12 Resource Masters - Math Problem Solving - …©Glencoe/McGraw-Hill v Glencoe Geometry Assessment Options The assessment masters in the Chapter 12 Resources Mastersoffer