Chapter 13 Resource Masters - Math Class©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 13 Resource Masters The Fast File Chapter Resource system allows

Chapter 13Resource 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 13 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: 0-07-860190-8 GeometryChapter 13 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 13-1Study Guide and Intervention . . . . . . . . 723–724Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 725Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 726Reading to Learn Mathematics . . . . . . . . . . 727Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 728





Chapter 13 AssessmentChapter 13 Test, Form 1 . . . . . . . . . . . 753–754Chapter 13 Test, Form 2A . . . . . . . . . . 755–756Chapter 13 Test, Form 2B . . . . . . . . . . 757–758Chapter 13 Test, Form 2C . . . . . . . . . . 759–760Chapter 13 Test, Form 2D . . . . . . . . . . 761–762Chapter 13 Test, Form 3 . . . . . . . . . . . 763–764Chapter 13 Open-Ended Assessment . . . . . 765Chapter 13 Vocabulary Test/Review . . . . . . 766Chapter 13 Quizzes 1 & 2 . . . . . . . . . . . . . . 767Chapter 13 Quizzes 3 & 4 . . . . . . . . . . . . . . 768Chapter 13 Mid-Chapter Test . . . . . . . . . . . . 769Chapter 13 Cumulative Review . . . . . . . . . . 770Chapter 13 Standardized Test Practice 771–772Unit 4 Test/Review (Ch. 4–7) . . . . . . . . 773–774Second Semester Test (Ch. 8–13) . . . . 775–778Final Test . . . . . . . . . . . . . . . . . . . . . . 779–784

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

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

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 13 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 13 Resource Masters includes the core materialsneeded for Chapter 13. 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 13-1.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 13Resources 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 724–725. 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 _____

1313

© 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 13. 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

congruent solids

ordered triple

similar solids

volume

Study Guide and InterventionVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

© Glencoe/McGraw-Hill 723 Glencoe Geometry

Less

on

13-

1

Volumes of Prisms The measure of the amount of space that a three-dimensional figure encloses is the volume of the figure. Volume is measured in units such as cubic feet, cubic yards, or cubic meters. One cubic unit is the volume of a cube that measures one unit on each edge.

27 cubic feet ! 1 cubic yard

Volume If a prism has a volume of V cubic units, a height of h units, of a Prism and each base has an area of B square units, then V ! Bh.

cubic foot cubic yard

Find the volume of the prism.

V ! Bh Formula for volume

! (7)(3)(4) B ! (7)(3), h ! 4

! 84 Multiply.The volume of the prism is 84 cubiccentimeters.

7 cm3 cm

4 cm

Find the volume of theprism if the area of each base is 6.3square feet.

V ! Bh Formula for volume

! (6.3)(3.5) B ! 6.3, h ! 3.5

! 22.05 Multiply.The volume is 22.05 cubic feet.

3.5 ft

base

Example 1Example 1 Example 2Example 2

ExercisesExercises

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

1. 2.

3. 4.

5. 6.

7 yd4 yd

3 yd

4 cm

6 cm

2 cm1.5 cm

10 ft15 ft

12 ft

30!15 ft

12 ft

3 cm4 cm

1.5 cm

8 ft

8 ft

8 ft


Volumes of Cylinders The volume of a cylinder is the product of the height and the area of the base. The base of a cylinder is a circle, so the area of the base is "r2.

Volume of If a cylinder has a volume of V cubic units, a height of h units, a Cylinder and the bases have radii of r units, then V ! "r 2h.

r

h

Study Guide and Intervention (continued)

Volumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

Find the volume of the cylinder.

V ! "r2h Volume of a cylinder

! "(3)2(4) r ! 3, h ! 4

! 113.1 Simplify.

The volume is about 113.1 cubiccentimeters.

4 cm

3 cm

Find the area of the oblique cylinder.

The radius of each base is 4 inches, so the area ofthe base is 16" in2. Use the Pythagorean Theoremto find the height of the cylinder.

h2 # 52 ! 132 Pythagorean Theorem

h2 ! 144 Simplify.

h ! 12 Take the square root of each side.

V ! "r2h Volume of a cylinder

! "(4)2(12) r ! 4, h !12

! 603.2 in3 Simplify.

8 in.

13 in.

5 in.

h

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the volume of each cylinder. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

1 yd4 yd

10 cm13 cm

20 ft

20 ft12 ft1.5 ft

18 cm2 cm2 ft

1 ft

Skills PracticeVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1


Less

on

13-

1

Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

Find the volume of each oblique prism or cylinder. Round to the nearest tenth ifnecessary.

7. 8.

5 in.

3 in.17 cm

18 cm

4 cm

6 yd

10 yd15 mm23 mm

16 in. 22 in.

34 in.

3 m

5 m

13 m

6 ft

8 ft

2 ft

18 cm

16 cm

8 cm


Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

AQUARIUM For Exercises 7–9, use the following information. Round answers tothe nearest tenth.Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25$

12$ inches and a radius of 21 inches.

7. What is the volume of the aquarium in cubic feet?

8. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquariumhold?

9. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in theaquarium to the nearest five pounds?

30 cm

8 cm

13 yd

20 yd

10 yd

7 ft 25 ft16 mm 17.5 mm

5 in.

5 in.

5 in.

9 in.17 m

10 m

26 m

Practice Volumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

Reading to Learn MathematicsVolumes of Prisms and Cylinders

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1


Less

on

13-

1

Pre-Activity How is mathematics used in comics?

Read the introduction to Lesson 13-1 at the top of page 688 in your textbook.

In the cartoon, why was Shoe confused when the teacher said the class wasgoing to discuss volumes?

Reading the Lesson1. In each case, write a formula for the volume V of the solid in terms of the given variables.

a. a rectangular box with length a, width b, and height c

b. a rectangular box with square bases with side length x, and with height y

c. a cube with edges of length e

d. a triangular prism whose bases are isosceles right triangles with legs of length x, andwhose height is y

e. a prism whose bases are regular polygons with perimeter P and apothem a, andwhose height is h

f. a cylinder whose bases each have radius r, and whose height is three times the radiusof the bases

g. a regular octagonal prism in which each base has sides of length s and apothem a,and whose height is t

h. a cylinder with height h whose bases each have diameter d

i. an oblique cylinder whose bases have radius a and whose height is b

j. a regular hexagonal prism whose bases have side length s, and whose height is h

Helping You Remember2. A good way to remember a mathematical concept is to explain it to someone else. Suppose

that your younger sister, who is in eighth grade, is having trouble understanding whysquare units are used to measure area, but cubic units are needed to measure volume.How can you explain this to her in a way that will make it easy for her to understandand remember the correct units to use?


Visible Surface Area

Use paper, scissors, and tape to make five cubes that have one-inch edges.Arrange the cubes to form each shape shown. Then find the volume and the visible surface area. In other words, do not include the area of surfacecovered by other cubes or by the table or desk.

1. 2.

volume ! volume !

surface area ! surface area !

3. 4. 5.

volume ! volume ! volume !

surface area ! surface area ! surface area !

6. Find the volume and the visible surface area of the figure at the right.

volume !

surface area !

4 in.

4 in.

3 in.

3 in.

3 in.

8 in.

3 in.

5 in.

5 in.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

13-113-1

Study Guide and InterventionVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


Less

on

13-

2

Volumes of Pyramids This figure shows a prism and a pyramid that have the same base and the same height. It is clear that the volume of the pyramid is less than the volume of the prism. More specifically,the volume of the pyramid is one-third of the volume of the prism.

Volume of If a pyramid has a volume of V cubic units, a height of h units, a Pyramid and a base with an area of B square units, then V ! $1

3$Bh.

Find the volume of the square pyramid.

V ! $13$Bh Volume of a pyramid

! $13$(8)(8)10 B ! (8)(8), h ! 10

! 213.3 Multiply.

The volume is about 213.3 cubic feet.

Find the volume of each pyramid. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6. 6 yd

8 yd

5 yd15 in.

15 in.

16 in.

18 ft

regularhexagon 6 ft

4 cm8 cm

12 cm

10 ft

6 ft15 ft

12 ft8 ft

10 ft

8 ft

8 ft

10 ft

ExercisesExercises

ExampleExample


Volumes of Cones For a cone, the volume is one-third the product of the height and the base. The base of a cone is a circle, so the area of the base is "r2.

Volume of a Right If a cone has a volume of V cubic units, a height of h units, Circular Cone and the area of the base is B square units, then V ! $1

3$Bh.

The same formula can be used to find the volume of oblique cones.

Find the volume of the cone.

V ! $13$"r2h Volume of a cone

! $13$"(5)212 r ! 5, h ! 12

! 314.2 Simplify.

The volume of the cone is about 314.2 cubic centimeters.

Find the volume of each cone. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.

16 cm

45!26 ft

20 ft

45!18 yd

20 yd30 in.

12 in.

8 ft

10 ft6 cm10 cm

12 cm

5 cm

r

h


Volumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2

ExercisesExercises

ExampleExample

Skills PracticeVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


Less

on

13-

2

Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

Find the volume of each oblique pyramid or cone. Round to the nearest tenth ifnecessary.

7. 8.

12 cm

6 cm

4 ft4 ft

6 ft

66!18 mm

25 yd

14 yd

25 m

12 m

8 in.10 in.

14 in.

4 cm7 cm

8 cm

5 ft5 ft

8 ft


Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. 6.

7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 metersin diameter, and the height of the shed is 3.8 meters. What is the volume of the shed?

8. HISTORY The start of the pyramid age began with King Zoser’s pyramid, erected in the27th century B.C. In its original state, it stood 62 meters high with a rectangular basethat measured 140 meters by 118 meters. Find the volume of the original pyramid.

37 ft11 ft

6 in.6 in.

11 in.

52!12 mm19 ft

9 ft

12.5 cm25 cm

23 cm

9.2 yd9.2 yd

13 yd

Practice Volumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2

Reading to Learn MathematicsVolumes of Pyramids and Cones

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2


Less

on

13-

2

Pre-Activity How do architects use geometry?


In addition to reflecting more light, why do you think the architect of theTransamerica Pyramid may have designed the building as a square pyramidrather than a rectangular prism?

Reading the Lesson1. In each case, two solids are described. Determine whether the first solid or the second

solid has the greater volume, or if the two solids have the same volume. (Answer bywriting first, second, or same.)a. First solid: A rectangular prism with length x, width y, and height z

Second solid: A rectangular prism with length 2x, width y, height zb. First solid: a rectangular prism that has a square base with side length x and that

has height ySecond solid: a square pyramid whose base has side length x and that has height y

c. First solid: a right cone whose base has radius x and that has height ySecond solid: an oblique cone whose base has radius x and that has height y

d. First solid: a cone whose base has radius x, and whose height is ySecond solid: a cylinder whose bases have radius x, and whose height is y

e. First solid: a cone whose base has radius x and whose height is ySecond solid: a square pyramid whose base has side length x and whose height is y

2. Supply the missing numbers to form true statements.a. If the length, width, and height of a rectangular box are all doubled, its volume will

be multiplied by .b. If the radius of a cylinder is tripled and the height is unchanged, the volume will be

multiplied by .c. In a square pyramid, if the side length of the base is multiplied by 1.5 and the height

is doubled, the volume will be multiplied by .d. In a cone, if the radius of the base is tripled and the height is doubled, the volume

will be multiplied by .e. In a cube, if the edge length is multiplied by 5, the volume will be multiplied by .

Helping You Remember

3. Many students find it easier to remember mathematical formulas if they can put themin words. Use words to describe in one sentence how to find the volume of any pyramidor cylinder.


FrustumsA frustum is a figure formed when a plane intersects a pyramid orcone so that the plane is parallel to the solid’s base. The frustum is the part of the solid between the plane and the base. To find thevolume of a frustum, the areas of both bases must be calculated andused in the formula

V ! $13

$h(B1 # B2 # "B1B2#),where h ! height (perpendicular distance between the bases),B1 ! area of top base, and B2 ! area of bottom base.

Describe the shape of the bases of each frustum. Then find the volume. Round to the nearest tenth.

1. 2.

3. 4.

12 ft13 ft

7 ft8 m

6 m

12 m

4.5 m2.25 m

3 m

5 m

3 in.

7.5 in.

4.5 in.

13 cm

6 cm

9 cm

5 cm

19.5 cm

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

13-213-2

Study Guide and InterventionVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3


Less

on

13-

3

Volumes of Spheres A sphere has one basic measurement, the length of its radius. If you know the radius of a sphere, you can calculate its volume.

Volume of a Sphere

If a sphere has a volume of V cubic units and a radius of r units, then V ! $43

$"r 3.

Find the volume of a sphere with radius 8 centimeters.

V ! $43$"r3 Volume of a sphere

! $43$"(8)3 r ! 8

! 2144.7 Simplify.

The volume is about 2144.7 cubic centimeters.

A sphere with radius 5 inches just fits inside a cylinder. What is the difference between the volume of thecylinder and the volume of the sphere? Round to the nearest cubic inch.The base of the cylinder is 25" in2 and the height is 10 in., so the volume of the cylinder is 250" in3. The volume of the sphere is $

43$"(5)3

or $5030"$ in3. The difference in the volumes is 250" % $

5030"$ or about 262 in3.

Find the volume of each solid. Round to the nearest tenth.

1. 2. 3.

4. 5. 6.

7. A hemisphere with radius 16 centimeters just fits inside a rectangular prism. What isthe difference between the volume of the prism and the volume of the hemisphere?Round to the nearest cubic centimeter.

8 in. difference between volume of cube and volume of sphere

13 in.5 in.

8 cm

16 in.

6 in.

5 ft

5 in.

5 in.5 in.

5 in.

8 cm

r

ExercisesExercises

Example 1Example 1

Example 2Example 2


Solve Problems Involving Volumes of Spheres If you want to know if a spherecan be packed inside another container, or if you want to compare the capacity of a sphereand another shape, you can compare volumes.

Compare the volumes of the sphere and the cylinder. Determine which quantity is greater.

V ! $43$"r3 Volume of sphere V ! "r2h Volume of cylinder

! "r2(1.5r) h ! 1.5r

! 1.5"r3 Simplify.

Compare $43$"r3 with 1.5"r3. Since $

43$ is less than 1.5, it follows that

the volume of the sphere is less than the volume of the cylinder.

Compare the volume of a sphere with radius r to the volume of each figure below.Which figure has a greater volume?

1. 2.

3. 4.

5. 6.2a

r

3r

r

r3r

r

rr

rr2

2r

r1.5r


Volumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3

ExercisesExercises

ExampleExample

Skills PracticeVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3


Less

on

13-

3

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

1. The radius of the sphere is 9 centimeters.

2. The diameter of the sphere is 10 inches.

3. The circumference of the sphere is 26 meters.

4. The radius of the hemisphere is 7 feet.

5. The diameter of the hemisphere is 12 kilometers.

6. The circumference of the hemisphere is 48 yards.

7. 8.

9. 10.

14.4 m

4.5 in.

94.8 ft16.2 cm


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

1. The radius of the sphere is 12.4 centimeters.

2. The diameter of the sphere is 17 feet.

3. The circumference of the sphere is 38 meters.

4. The diameter of the hemisphere is 21 inches.

5. The circumference of the hemisphere is 18 millimeters.

6. 7.

8. 9.

10. PACKAGING Amber plans to ship a mini-basketball she bought for her nephew. Thecircumference of the ball is 24 inches and the package she wants to ship it in is arectangular box that measures 8 inches & 8 inches & 9 inches. Will the basketball fit inthe box? Explain.

C " 43 mm

32 m

C " 58 cm12.32 ft

Practice Volumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3

Reading to Learn MathematicsVolumes of Spheres

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3


Less

on

13-

3

Pre-Activity How can you find the volume of Earth?


How would you estimate the radius of Earth based on Eratosthenes’estimate of its diameter?

Reading the Lesson

1. Name all solids from the following list for which each volume formula can be used:prism, pyramid, cone, cylinder, sphere, hemisphere.

a. V ! Bh b. V ! $43$"r3

c. V ! $13$Bh d. V ! "r2h

e. V ! $13$"r2h f. V ! $

23$"r3

2. Let r represent the radius and d represent the diameter of a sphere. Determine whethereach formula below can be used to find the volume of a sphere, a hemisphere, or neither.

a. V ! $2"

3r3$ b. V ! $

16$"d3

c. V ! $13$"r3 d. V ! $

34$"r3

e. V ! $"1d2

3$ f. V ! $

43$"r2h

3. Compare the volumes of these three solids. Then complete the sentence below.

Of the three solids shown above, the has the largest volume and the

has the smallest volume.


4. A good way to remember something is to explain it to someone else. Suppose that your classmate Loretta knows that the expressions $

43$"r3 and 4"r2 are used in finding

measurements related to spheres, but can’t remember which one is used to find thesurface area of a sphere and which one is used to find the volume. How can you help herto remember which is which?

2r

r

rrr


Spheres and DensityThe density of a metal is a ratio of its mass to its volume. Forexample, the mass of aluminum is 2.7 grams per cubic centimeter.Here is a list of several metals and their densities.

Aluminum 2.7 g/cm3 Copper 8.96 g/cm3

Gold 19.32 g/cm3 Iron 7.874 g/cm3

Lead 11.35 g/cm3 Platinum 21.45 g/cm3

Silver 10.50 g/cm3

To calculate the mass of a piece of metal, multiply volume by density.

Find the mass of a silver ball that is 0.8 cm in diameter.

M ! D ' V

! 10.5 ' $43$"(0.4)3

! 10.5 (0.27)! 2.83

The mass is about 2.83 grams.

Find the mass of each metal ball described. Assume the balls are spherical. Round your answers to the nearest tenth.

1. a copper ball 1.2 cm in diameter

2. a gold ball 0.6 cm in diameter

3. an aluminum ball with radius 3 cm

4. a platinum ball with radius 0.7 cm

Solve. Assume the balls are spherical. Round your answers to the nearest tenth.

5. A lead ball weighs 326 g. Find the radius of the ball to the nearest tenth of a centimeter.

6. An iron ball weighs 804 g. Find the diameter of the ball to the nearest tenth of a centimeter.

7. A silver ball and a copper ball each have a diameter of 3.5 cm.Which weighs more? How much more?

8. An aluminum ball and a lead ball each have a radius of 1.2 cm.Which weighs more? How much more?

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

13-313-3

ExampleExample

Study Guide and InterventionCongruent and Similar Solids

NAME ______________________________________________ DATE ____________ PERIOD _____

13-413-4


Less

on

13-

4

Congruent or Similar Solids If the corresponding angles and sides of two solids arecongruent, then the solids are congruent. Also, the corresponding faces are congruent andtheir surface areas and volumes are equal. Solids that have the same shape but aredifferent sizes are similar. You can determine whether two solids are similar by comparingthe ratio, or scale factor, of corresponding linear measurements.

Describe each pair of solids.

• Figures I and II are similar because the figures have the same shape. The ratio of eachpair of corresponding sides is 1:3.

• Figures III and IV are congruent because they have the same shape and all correspondingmeasurements are the same.

• Figures V and VI are not congruent, and they are not similar because $48$ ( $

1122$.

Determine whether each pair of solids are similar, congruent, or neither.

1. 2.

3. 4.

5. 6.2

7

21

6

5

8

5

8

4

4

88

5

5

2

2

2

26

6

7

7

12

4

5

1

106

8

53

4

I II III IV V VIsimilar congruent non-similar

12 5

5

5 12

12 4

85

5

5

7

7

9

6

4 32

ExercisesExercises

ExampleExample


Properties of Similar Solids These two solids are similar with a scale factor of 1:2. The surface areas are 62 cm2 and 248 cm2 and the volumes are 30 cm3 and 240 cm3. Notice that the ratio of the surface areas is 62:248, which is 1:4 or 12:22, and the ratio of the volumes is 30:240, which is 1:8 or 13:23.

If two solids are similar with a scale factor of a :b, then the surface areas have a ratio of a2:b2, and the volumes have a ratio of a3:b3.

Use the two spheres.a. Find the scale factor for the two spheres.

The scale factor for the two spheres is the same as the ratio of their radii, or 5:3.

b. Find the ratio of the surface areas of the two spheres.The ratio of the surface areas is 52:32 or 25:9.

c. Find the ratio of the volumes of the two spheres.The ratio of the volumes is 53:33 or 125:27.

Find the scale factor for each pair of similar figures. Then find the ratio of theirsurface areas and the ratio of their volumes.

1. 2.

3. 4.

5. 6.

8 65

3

1215

4 yd16 yd

15 m12 m

7 in. 4 in.

3 ft4 ft

5 cm3 cm

10 cm5 cm2 cm

3 cm

6 cm

4 cm


Congruent and Similar Solids

NAME ______________________________________________ DATE ____________ PERIOD _____

13-413-4

ExercisesExercises

ExampleExample

Skills PracticeCongruent and Similar Solids

NAME ______________________________________________ DATE ____________ PERIOD _____

13-413-4


Less

on

13-

4


1.

2.

3.

4.

For Exercises 5–8, refer to the two similar prisms.

5. Find the scale factor of the two prisms.

6. Find the ratio of the surface areas.

7. Find the ratio of the volumes.

8. Suppose the volume of the larger prism is 810 cubic centimeters. Find the volume of thesmaller prism.

15 cm12 cm

9 cm 10 cm8 cm

6 cm

18 in.

16 in.16 in.

9 in.

6 mm

6 mm

4 mm9 mm

12 ft12 ft

14 ft

20 ft20 ft

21 ft

20 cm20 cm

10 cm

4 cm40 cm

8 cm



1.

2.

3.

4.

For Exercises 5–8, refer to the two similar prisms.

5. Find the scale factor of the two prisms.

6. Find the ratio of the surface areas.

7. Find the ratio of the volumes.

8. Suppose the surface area of the larger prism is 2560 square meters. Find the surfacearea of the smaller prism.

9. MINIATURES Frank Lloyd Wright designed every aspect of the Imperial Hotel in Tokyo,including the chairs. The dimensions of a miniature Imperial Hotel chair are 6.25 inches &3 inches & 2.5 inches. If the scale of the replica is 1:6, what are the dimensions of theoriginal chair?

20 m

20 m

22 m12 m

12 m

13.2 m

7.5 cm

20 cm

15 cm

4.5 cm

12 cm9 cm

18 ft24 ft

24 ft9 ft

12 m

12 m15 m

2.5 m

2 m 9.6 m

25 in.

15 in.

30 in.20 in.

Practice Congruent and Similar Solids

NAME ______________________________________________ DATE ____________ PERIOD _____

13-413-4

Reading to Learn MathematicsCongruent and Similar Solids

NAME ______________________________________________ DATE ____________ PERIOD _____

13-413-4


Less

on

13-

4

Pre-Activity How are similar solids applied to miniature collectibles?


If you want to make a miniature with a scale factor of 1:64, how can youuse the actual object to find the measurements you should use to constructthe miniature?

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

a. Two cubes are similar.b. Two cones are similar.c. Two cylinders in which the height is twice the diameter are similar.d. Two cylinders with the same volume are congruent.e. A prism with a square base and a square pyramid are similar.f. Two rectangular prisms with equal surface areas are similar.g. Nonsimilar solids have different volumes.h. Two hemispheres with the same radius are congruent.

2. Supply the missing ratios.

a. If the ratio of the diameters of two spheres is 3:1, then the ratio of their surface areas

is , and the ratio of their volumes is .

b. If the ratio of the radii of two hemispheres is 2:5, then the ratio of their surface areas

is , and the ratio of their volumes is .

c. If two cones are similar and the ratio of their heights is $43$, then the ratio of their

volumes is , and the ratio of their surface areas is .

d. If two cylinders are similar and the ratio of their surface areas is 100:49, then the

ratio of the radii of their bases is , and the ratio of their volumes is

.

Helping You Remember3. A good way to remember a new mathematical concept is to relate it to something you

already know. How can what you know about the units used to measure lengths, areas,and volumes help you to remember the theorem about the ratios of surface areas andvolumes of similar solids?


Congruent and Similar Solids

Determine whether each pair of solids is similar, congruent, or neither.

1. 2.

3. 4.

The two rectangular prisms shown at the right are similar.

5. Find the ratio of the perimeters of the bases.

6. What is the ratio of the surface areas?

7. Suppose the volume of the smaller prism is 60 in3.Find the volume of the larger prism.

Determine whether each statement is true or false. If the statement is false, rewrite it so that it is true.

8. If two cylinders are similar, then their volumes are equal.

9. Doubling the height of a cylinder doubles the volume.

10. Two solids are congruent if they have the same shape.

7 in.5 in.

24 yd

12 yd

12 yd6 yd

8 yd

16 yd12 m

3 m

3 m

3 m

3 m

3 m3 m

4 m

4 m4 m 4 m

4 m

10 m

48 m

16 m

15 m

14 cm

11 cm

7 cm

7 cm

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

13-413-4

Study Guide and InterventionCoordinates in Space

NAME ______________________________________________ DATE ____________ PERIOD _____

13-513-5


Less

on

13-

5

Graph Solids in Space In space, you can describe the location of a point using an ordered triple of realnumbers. The x-, y-, and z-axes are perpendicular to each other, and the coordinates for point P are the ordered triple (%4, 6, 5). A rectangular prism can bedrawn to show perspective.

Graph the rectangular solid that contains the ordered triple (2, 1, %2) and the origin. Label the coordinates of each vertex.• Plot the x-coordinate first. Draw a solid segment

from the origin 2 units in the positive direction.• Plot the y-coordinate next. Draw a solid segment

1 unit in the positive direction.• Plot the z-coordinate next. Draw a solid segment

2 units in the negative direction.• Draw the rectangular prism, using dotted lines for

hidden edges of the prism.• Label the coordinates of each vertex.

Graph the rectangular solid that contains the given point and the origin asvertices. Label the coordinates of each vertex.

1. A(2, 1, 3) 2. G(%1, 2, 3)

3. P(%2, 1, %1) 4. T(%1, 3, 2)

y

x

z

(0, 0, 0)

(0, 3, 0)

(#1, 3, 0)

(0, 0, 2) (0, 3, 2)

(#1, 0, 2)

(#1, 0, 0)

T (#1, 3, 2)

y

x

z

(0, 0, 0)

(0, 1, 0)

P(#2, 1, #1)

(#2, 0, #1) (#2, 1, 0)(#2, 0, 0)

(0, 0, #1) (0, 1, #1)

y

x

z

(0, 0, 0)

(0, 0, 3)

(#1, 2, 0)(#1, 0, 0)

(#1, 0, 3)G(#1, 2, 3)

(0, 2, 0)

(0, 2, 3)

y

x

z(0, 0, 3)

(0, 0, 0)

(2, 0, 3)

(2, 0, 0)(2, 1, 0)

(0, 1, 0)

(0, 1, 3)

A(2, 1, 3)

y

x

z

(0, 0, 0) (0, 1, 0)

(0, 1, #2)

(2, 1, #2)(2, 0, #2)

(0, 0, #2)

(2, 0, 0) (2, 1, 0)

y

x

z

O

P(#4, 6, 5)

ExercisesExercises

ExampleExample


Distance and Midpoint Formulas You can extend the Distance Formula and theMidpoint Formula to three dimensions to find the distance between two points in space and to find the midpoint of the segment connecting two points.

Distance Formula Given two points A(x1, y1, z1) and B(x2, y2, z2) in space, the distance between

in Space A and B is given by AB ! "(x1 %#x2)2 ## (y1 %# y2)2 ## (z1 %# z2)2#.

Midpoint Formula Given two points A(x1, y1, z1) and B(x2, y2, z2) in space, the midpoint of A#B# is

in Space at $$x1 #

2x2$, $

y1 #

2y2$, $

z1 #

2z2$%.

Determine the distance between A(3, 2, !5) and B(%4, 6, 9).Then determine the coordinates of the midpoint of A#B#.

AB ! "(x1 %#x2)2 ## ( y1 %# y2)2 ## (z1 %# z2)2#! "(3 % (#%4))2## (2 %# 6)2 ## (%5 %# 9)2#! "72 # (#%4)2 ## (%14#)2#! "49 ##16 ##196#! 16.2

midpoint of A#B# ! $$x1 #

2x2$, $

y1 #

2y2$, $

z1 #

2z2$%

! $$3 #2(%4)$, $

2 #2

6$, $%5

2# 9$%

! (%0.5, 4, 2)

Determine the distance between each pair of points. Then determine thecoordinates of the midpoint M of the segment joining the pair of points.

1. A(0, 7, %4) and B(%2, 8, 3) 2. C(%7, 6, 5) and D(10, 2, %5)

3. E(3, 1, %2) and F(%2, 3, 4) 4. G(%4, 1, 1) and H(0, 2, %1)

5. J(6, 1, %2) and K(%1, %2, 1) 6. L(%5, 0, %3) and N(0, 0, %4)


Coordinates in Space

NAME ______________________________________________ DATE ____________ PERIOD _____

13-513-5

ExercisesExercises

ExampleExample

Skills PracticeCoordinates in Space

NAME ______________________________________________ DATE ____________ PERIOD _____

13-513-5


Less

on

13-

5


1. A(%5, 3, 2) 2. H(3, 2, 5)

3. Dilate the prism by a scale factor of 2. Graph the image under the dilation.


4. R(2, 1, 0) and S(3, 3, 4) 5. Q(5, 0, %2) and T(2, 3, 2)

6. A(%4, 1, 6) and B(%1, 0, 4) 7. J(0, 5, 1) and K(4, %3, 2)

A$B$

C$ D$E$F$

G$H$

y

x

z

AB

C

G H

ED

F

N(0, 0, 0)

M(3, 0, 0)L(3, 2, 0)

P(0, 2, 0)

K(0, 2, 5)J(0, 0, 5)

I(3, 0, 5)

H(3, 2, 5)

A(#5, 3, 2)

E(#5, 3, 0)

B(#5, 0, 2)

C(0, 0, 2)D(0, 3, 2)

H(0, 3, 0)

G(0, 0, 0)

F (#5, 0, 0)



1. E(4, 6, %2) 2. R(%3, %5, 4)


3. Y(%5, 1, 2) and Z(3, %3, 1) 4. E(4, 2, 0) and F(3, 2, %2)

5. B(%2, %2, %3) and C(1, %3, 0) 6. H(2, 0, %3) and I(4, %1, 5)

7. ANIMATION Derek wants to animate an image for his science presentation by movingit from one position to another. The mesh of the image is a rectangular prism withcoordinates A(%3, 2, 3), B(%3, 0, 3), C(0, 0, 3), D(0, 2, 3), E(%3, 2, 0), F(%3, 0, 0), G(0, 0, 0),and H(0, 2, 0). Find the coordinates of the mesh after the translation (x, y, z) → (x % 7, y, z).Graph both the preimage and image of the mesh.

A$B$

C$ D$

E$

F$G$ H$

AB

C D EFG

H

X(0, 0, 0)

Y(0, #5, 0)

V(#3, #5, 0)

R(#3, #5, 4) S(#3, 0, 4)

T(0, 0, 4)U(0, #5, 4)

W(#3, 0, 0)

K(0, 0, 0)

J(4, 0, 0)I(4, 6, 0)

L(0, 6, 0)

H(0, 6, #2)

E(4, 6, #2)F(4, 0, #2)

G(0, 0, #2)

Practice Coordinates in Space

NAME ______________________________________________ DATE ____________ PERIOD _____

13-513-5

Reading to Learn MathematicsCoordinates in Space

NAME ______________________________________________ DATE ____________ PERIOD _____

13-513-5


Less

on

13-

5

Pre-Activity How is three-dimensional graphing used in computer animation?


Why would a mesh be created first?

Reading the Lesson

1. Refer to the figure. Match each point from the first column with its coordinates from the second column.

a. A i. (3, 0, 0)

b. B ii. (3, 0, %4)

c. O iii. (3, %2, 0)

d. J iv. (3, %2, %4)

e. H v. (0, 0, 0)

f. K vi. (0, %2, 0)

g. T vii. (0, %2, %4)

h. R viii. (0, 0, %4)

2. Which of the following expressions give the distance between the points at (4, %1, %5)and (%3, 2, %9)?

A. "72 # (#%3)2 ## 42# B. "12 # 1#2 # (%#14)2#

C. "22 # 2#2 # 42# D. $$12$, $

12$, %7%

E. "(%3 %# 4)2 ## (%1 %# 2)2 ## (%9 ## 5)2# F. "24#

G. "(%3 ## 4)2 ## [2 ##(%1)]2# # [%#9 # (%#5)]2# H. "74#


3. A good way to remember new mathematical formulas is to relate them to ones youalready know. How can you use your knowledge of the Distance and Midpoint Formulasin two dimensions to remember the formulas in three dimensions?

y

x

z

A

K

BO

R

H

T

J


Planes and Cylindrical SurfacesConsider the points (x, y, z) in space whose coordinates satisfy the equation z ! 1. Since x and y do not occur in the equation, any point with its z-coordinate equal to 1 has coordinates that satisfy the equation. These are the points in the plane 1 unit above the xy-plane. This plane is perpendicular to the z-axis at (0, 0, 1).

Next consider the points (x, y, z) whose coordinates satisfy x2 # y2 ! 16. In the xy-plane,all points on the circle with center (0, 0, 0) andradius 4 have coordinates that satisfy the equation. In the plane perpendicular to the z-axis at (0, 0, k), the points that satisfy theequation are those on the circle with center (0, 0, k) and radius 4. The graph in space of x2 # y2 ! 16 is an infinite cylindrical surface whose axis is the z-axis and whose radius is 4.

Describe the graph in space of each equation. You may find it helpful to make sketches on a separate sheet.

1. x ! 5

2. y ! %2

3. x # y ! 7

4. z2 # y2 ! 25

5. (x % 2)2 # (y % 5)2 ! 1

6. x2 # y2 # z2 ! 0

z

y

x

O

(0, 0, k)

plane for z " k

z

y

x

O

(0, 0, 1)N

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

13-513-5

© Glencoe/McGraw-Hill A2 Glencoe Geometry

Stu

dy

Gu

ide

and I

nte

rven

tion

Volu

mes

of P

rism

s an

d C

ylin

ders

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-1

13-1

©G

lenc

oe/M

cGra

w-H

ill72

3G

lenc

oe G

eom

etry

Lesson 13-1

Vo

lum

es o

f Pr

ism

sT

he m

easu

re o

f th

e am

ount

of

spac

e th

at a

thr

ee-d

imen

sion

al f

igur

e en

clos

es is

the

vol

um

eof

the

fi

gure

.Vol

ume

is m

easu

red

in u

nits

suc

h as

cub

ic f

eet,

cubi

c ya

rds,

or c

ubic

met

ers.

One

cub

ic u

nit

is t

he v

olum

e of

a c

ube

that

mea

sure

s on

e un

it o

n ea

ch e

dge.

27 c

ubic

feet

!1

cubi

c ya

rd

Volu

me

If a

pris

m h

as a

vol

ume

of V

cubi

c un

its, a

hei

ght o

f hun

its,

of a

Pri

sman

d ea

ch b

ase

has

an a

rea

of B

squa

re u

nits

, the

n V

!B

h.

cubi

c fo

otcu

bic

yard

Fin

d t

he

volu

me

of t

he

pri

sm.

V!

Bh

For

mul

a fo

r vo

lum

e

!(7

)(3)

(4)

B!

(7)(

3), h

!4

!84

Mul

tiply.

The

vol

ume

of t

he p

rism

is 8

4 cu

bic

cent

imet

ers.

7 cm

3 cm4

cm

Fin

d t

he

volu

me

of t

he

pri

sm i

f th

e ar

ea o

f ea

ch b

ase

is 6

.3sq

uar

e fe

et.

V!

Bh

For

mul

a fo

r vo

lum

e

!(6

.3)(

3.5)

B!

6.3,

h!

3.5

!22

.05

Mul

tiply.

The

vol

ume

is 2

2.05

cub

ic f

eet.

3.5

ft

base

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

volu

me

of e

ach

pri

sm.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

512

ft3

9 cm

3

3.4.

467.

7 ft

318

00 f

t3

5.6.

27 c

m3

84 y

d37 yd

4 yd

3 yd

4 cm

6 cm

2 cm

1.5

cm

10 ft

15 ft

12 ft

30!

15 ft12

ft

3 cm

4 cm

1.5

cm

8 ft

8 ft

8 ft

©G

lenc

oe/M

cGra

w-H

ill72

4G

lenc

oe G

eom

etry

Vo

lum

es o

f C

ylin

der

sT

he v

olum

e of

a c

ylin

der

is t

he p

rodu

ct o

f th

e he

ight

and

the

are

a of

the

bas

e.T

he b

ase

of a

cyl

inde

r is

a c

ircl

e,so

the

are

a of

the

bas

e is

"r2

.

Volu

me

of

If a

cylin

der

has

a vo

lum

e of

Vcu

bic

units

, a h

eigh

t of h

units

, a

Cyl

inde

ran

d th

e ba

ses

have

rad

ii of

run

its, t

hen

V!

"r2

h.

r

h

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Volu

mes

of P

rism

s an

d C

ylin

ders

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-1

13-1

Fin

d t

he

volu

me

of t

he

cyli

nd

er.

V!

"r2

hV

olum

e of

a c

ylin

der

!"

(3)2

(4)

r!

3, h

!4

!11

3.1

Sim

plify

.

The

vol

ume

is a

bout

113

.1 c

ubic

cent

imet

ers.4

cm3 cm

Fin

d t

he

area

of

the

obli

que

cyli

nd

er.

The

rad

ius

of e

ach

base

is 4

inch

es,s

o th

e ar

ea o

fth

e ba

se is

16"

in2 .

Use

the

Pyt

hago

rean

The

orem

to f

ind

the

heig

ht o

f th

e cy

linde

r.

h2#

52!

132

Pyt

hago

rean

The

orem

h2!

144

Sim

plify

.

h!

12Ta

ke th

e sq

uare

roo

t of e

ach

side

.

V!

"r2

hV

olum

e of

a c

ylin

der

!"

(4)2

(12)

r!

4, h

!12

!60

3.2

in3

Sim

plify

.

8 in

.13 in

.

5 in

.

h

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

volu

me

of e

ach

cyl

ind

er.R

oun

d t

o th

e n

eare

st t

enth

.

1.2.

12.6

ft3

226.

2 cm

3

3.4.

84.8

ft3

6283

.2 f

t3

5.6.

652.

4 cm

312

.6 y

d3

1 yd

4 yd

10 c

m13

cm

20 ft

20 ft

12 ft

1.5

ft

18 c

m2

cm2

ft

1 ft

Answers (Lesson 13-1)


An

swer

s

Skil

ls P

ract

ice

Volu

mes

of P

rism

s an

d C

ylin

ders

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-1

13-1

©G

lenc

oe/M

cGra

w-H

ill72

5G

lenc

oe G

eom

etry

Lesson 13-1

Fin

d t

he

volu

me

of e

ach

pri

sm o

r cy

lin

der

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

1.2.

2304

cm

396

ft3

3.4.

90 m

352

80 in

3

5.6.

16,2

57.7

mm

322

6.2

yd3

Fin

d t

he

volu

me

of e

ach

obl

iqu

e p

rism

or

cyli

nd

er.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

7.8.

1224

cm

3

141.

4 in

3

5 in

.

3 in

.17

cm

18 c

m

4 cm

6 yd 10

yd

15 m

m23

mm

16 in

.22

in.

34 in

.

3 m

5 m

13 m

6 ft

8 ft

2 ft

18 c

m

16 c

m8 cm

©G

lenc

oe/M

cGra

w-H

ill72

6G

lenc

oe G

eom

etry

Fin

d t

he

volu

me

of e

ach

pri

sm o

r cy

lin

der

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

1.2.

2040

m3

97.4

in3

3.4.

3518

.6 m

m3

923.

6 ft

3

5.6.

2600

yd3

6031

.9 c

m3

AQ

UA

RIU

MF

or E

xerc

ises

7–9

,use

th

e fo

llow

ing

info

rmat

ion

.Rou

nd

an

swer

s to

the

nea

rest

ten

th.

Mr.

Gut

ierr

ez p

urch

ased

a c

ylin

dric

al a

quar

ium

for

his

off

ice.

The

aqu

ariu

m h

as a

hei

ght

of

25$1 2$

inch

es a

nd a

rad

ius

of 2

1 in

ches

.

7.W

hat

is t

he v

olum

e of

the

aqu

ariu

m in

cub

ic f

eet?

5.1

ft3

8.If

the

re a

re 7

.48

gallo

ns in

a c

ubic

foo

t,ho

w m

any

gallo

ns o

f w

ater

doe

s th

e aq

uari

umho

ld?

38.2

gal

9.If

a c

ubic

foo

t of

wat

er w

eigh

s ab

out

62.4

pou

nds,

wha

t is

the

wei

ght

of t

he w

ater

in t

heaq

uari

um t

o th

e ne

ares

t fi

ve p

ound

s?

2385

lb

30 c

m

8 cm

13 y

d

20 y

d

10 y

d

7 ft

25 ft

16 m

m17

.5 m

m

5 in

.

5 in

.

5 in

.

9 in

.17

m10

m

26 mP

ract

ice

(Ave

rage

)

Volu

mes

of P

rism

s an

d C

ylin

ders

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-1

13-1



Rea

din

g t

o L

earn

Math

emati

csVo

lum

es o

f Pri

sms

and

Cyl

inde

rs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-1

13-1

©G

lenc

oe/M

cGra

w-H

ill72

7G

lenc

oe G

eom

etry

Lesson 13-1

Pre-

Act

ivit

yH

ow i

s m

ath

emat

ics

use

d i

n c

omic

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-1 a

t th

e to

p of

pag

e 68

8 in

you

r te

xtbo

ok.

In t

he c

arto

on,w

hy w

as S

hoe

conf

used

whe

n th

e te

ache

r sa

id t

he c

lass

was

goin

g to

dis

cuss

vol

umes

?S

ampl

e an

swer

:The

tea

cher

was

refe

rrin

g to

the

mat

hem

atic

al m

eani

ng o

f vo

lum

e,w

hich

is t

heam

ount

of

spac

e en

clos

ed b

y a

thre

e-di

men

sion

al f

igur

e.S

hoe

was

thi

nkin

g ab

out

volu

mes

as

book

s,w

hich

is a

com

plet

ely

diff

eren

t m

eani

ng o

f th

e w

ord.

Rea

din

g t

he

Less

on

1.In

eac

h ca

se,w

rite

a fo

rmul

a fo

r th

e vo

lum

e V

of t

he s

olid

in t

erm

s of

the

giv

en v

aria

bles

.

a.a

rect

angu

lar

box

wit

h le

ngth

a,w

idth

b,a

nd h

eigh

t c

V"

abc

b.a

rect

angu

lar

box

wit

h sq

uare

bas

es w

ith

side

leng

th x

,and

wit

h he

ight

yV

"x2

yc.

a cu

be w

ith

edge

s of

leng

th e

V"

e3

d.a

tria

ngul

ar p

rism

who

se b

ases

are

isos

cele

s ri

ght

tria

ngle

s w

ith

legs

of

leng

th x

,and

who

se h

eigh

t is

yV

"#1 2# x

2 yor

V"

#x 22 y #e.

a pr

ism

who

se b

ases

are

reg

ular

pol

ygon

s w

ith

peri

met

er P

and

apot

hem

a,a

ndw

hose

hei

ght

is h

V"

#1 2# aP

hor

V"

#aP 2h #

f.a

cylin

der

who

se b

ases

eac

h ha

ve r

adiu

s r,

and

who

se h

eigh

t is

thr

ee t

imes

the

rad

ius

of t

he b

ases

V"

3$r3

g.a

regu

lar

octa

gona

l pri

sm in

whi

ch e

ach

base

has

sid

es o

f le

ngth

san

d ap

othe

m a

,an

d w

hose

hei

ght

is t

V"

4ast

h.

a cy

linde

r w

ith

heig

ht h

who

se b

ases

eac

h ha

ve d

iam

eter

d

V"

$!#d 2# "2 h

or V

"#$

d 42 h #or

V"

#1 4# $d

2 h

i.an

obl

ique

cyl

inde

r w

hose

bas

es h

ave

radi

us a

and

who

se h

eigh

t is

bV

"$

a2b

j.a

regu

lar

hexa

gona

l pri

sm w

hose

bas

es h

ave

side

leng

th s

,and

who

se h

eigh

t is

h

V"

#3 2# s2 h

#3$

Hel

pin

g Y

ou

Rem

emb

er2.

A g

ood

way

to

rem

embe

r a

mat

hem

atic

al c

once

pt is

to

expl

ain

it t

o so

meo

ne e

lse.

Supp

ose

that

you

r yo

unge

r si

ster

,who

is in

eig

hth

grad

e,is

hav

ing

trou

ble

unde

rsta

ndin

g w

hysq

uare

uni

ts a

re u

sed

to m

easu

re a

rea,

but

cubi

c un

its

are

need

ed t

o m

easu

re v

olum

e.H

ow c

an y

ou e

xpla

in t

his

to h

er in

a w

ay t

hat

will

mak

e it

eas

y fo

r he

r to

und

erst

and

and

rem

embe

r th

e co

rrec

t un

its

to u

se?

Sam

ple

answ

er:A

rea

mea

sure

s th

eam

ount

of

spac

e in

side

a t

wo-

dim

ensi

onal

fig

ure,

whi

le v

olum

e m

easu

res

the

amou

nt o

f sp

ace

insi

de a

thr

ee-d

imen

sion

al f

igur

e.A

tw

o-di

men

sion

alfig

ure

can

be c

over

ed w

ith s

mal

l squ

ares

,whi

ch r

epre

sent

squ

are

units

,w

hile

a t

hree

-dim

ensi

onal

fig

ure

can

be f

illed

with

sm

all c

ubes

,whi

chre

pres

ent

cubi

c un

its.

©G

lenc

oe/M

cGra

w-H

ill72

8G

lenc

oe G

eom

etry

Vis

ible

Sur

face

Are

a

Use

pap

er,s

ciss

ors,

and

tap

e to

mak

e fi

ve c

ube

s th

at h

ave

one-

inch

ed

ges.

Arr

ange

th

e cu

bes

to f

orm

eac

h s

hap

e sh

own

.Th

en f

ind

th

e vo

lum

e an

d

the

visi

ble

surf

ace

area

.In

oth

er w

ord

s,d

o n

ot i

ncl

ud

e th

e ar

ea o

f su

rfac

eco

vere

d b

y ot

her

cu

bes

or b

y th

e ta

ble

or d

esk

.

1.2.

volu

me

!4

in3

volu

me

!4

in3

surf

ace

area

!14

in2

surf

ace

area

!15

in2

3.4.

5.

volu

me

!5

in3

volu

me

!5

in3

volu

me

!5

in3

surf

ace

area

!17

in2

surf

ace

area

!19

in2

surf

ace

area

!19

in2

6.F

ind

the

volu

me

and

the

visi

ble

surf

ace

area

of

the

figu

re a

t th

e ri

ght.

volu

me

!12

4 in

3

surf

ace

area

!15

8 in

2

4 in

.

4 in

.

3 in

.

3 in

.

3 in

.

8 in

.

3 in

. 5 in

.

5 in

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-1

13-1



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Volu

mes

of P

yram

ids

and

Con

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-2

13-2

©G

lenc

oe/M

cGra

w-H

ill72

9G

lenc

oe G

eom

etry

Lesson 13-2

Vo

lum

es o

f Py

ram

ids

Thi

s fi

gure

sho

ws

a pr

ism

and

a p

yram

id

that

hav

e th

e sa

me

base

and

the

sam

e he

ight

.It

is c

lear

tha

t th

e vo

lum

e of

the

pyr

amid

is le

ss t

han

the

volu

me

of t

he p

rism

.Mor

e sp

ecif

ical

ly,

the

volu

me

of t

he p

yram

id is

one

-thi

rd o

f th

e vo

lum

e of

the

pri

sm.

Volu

me

ofIf

a py

ram

id h

as a

vol

ume

of V

cubi

c un

its, a

hei

ght o

f hun

its,

a P

yram

idan

d a

base

with

an

area

of B

squa

re u

nits

, the

n V

!$1 3$ B

h.

Fin

d t

he

volu

me

of t

he

squ

are

pyr

amid

.

V!

$1 3$ Bh

Vol

ume

of a

pyr

amid

! $1 3$ (

8)(8

)10

B!

(8)(

8), h

!10

!21

3.3

Mul

tiply

.

The

vol

ume

is a

bout

213

.3 c

ubic

fee

t.

Fin

d t

he

volu

me

of e

ach

pyr

amid

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

1.2.

320

ft3

120

ft3

3.4.

110.

9 cm

356

1.2

ft3

5.6.

1200

in3

64 y

d3

6 yd

8 yd

5 yd

15 in

.

15 in

.

16 in

.

18 ft

regu

lar

hexa

gon

6 ft

4 cm

8 cm

12 c

m

10 ft

6 ft

15 ft

12 ft

8 ft

10 ft

8 ft

8 ft

10 ft

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill73

0G

lenc

oe G

eom

etry

Vo

lum

es o

f C

on

esFo

r a

cone

,the

vol

ume

is o

ne-t

hird

the

pro

duct

of

the

heig

ht a

nd t

he b

ase.

The

bas

e of

a c

one

is a

cir

cle,

so t

he a

rea

of t

he b

ase

is "

r2.

Volu

me

of a

Rig

ht

If a

cone

has

a v

olum

e of

Vcu

bic

units

, a h

eigh

t of h

units

, C

ircu

lar

Con

ean

d th

e ar

ea o

f the

bas

e is

Bsq

uare

uni

ts, t

hen

V!

$1 3$ Bh.

The

sam

e fo

rmul

a ca

n be

use

d to

fin

d th

e vo

lum

e of

obl

ique

con

es.

Fin

d t

he

volu

me

of t

he

con

e.

V!

$1 3$ "r2

hV

olum

e of

a c

one

! $1 3$ "

(5)2

12r

!5,

h!

12

!31

4.2

Sim

plify

.

The

vol

ume

of t

he c

one

is a

bout

314

.2 c

ubic

cen

tim

eter

s.

Fin

d t

he

volu

me

of e

ach

con

e.R

oun

d t

o th

e n

eare

st t

enth

.

1.2.

301.

6 cm

367

0.2

ft3

3.4.

1131

.0 in

313

32.9

yd3

5.6.

2513

.3 f

t337

9.1

cm3

16 c

m

45!

26 ft

20 ft

45!

18 y

d 20 y

d30

in.

12 in

.

8 ft

10 ft

6 cm

10 c

m

12 c

m

5 cm

r

h

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Volu

mes

of P

yram

ids

and

Con

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-2

13-2

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Volu

mes

of P

yram

ids

and

Con

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-2

13-2

©G

lenc

oe/M

cGra

w-H

ill73

1G

lenc

oe G

eom

etry

Lesson 13-2

Fin

d t

he

volu

me

of e

ach

pyr

amid

or

con

e.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

66.7

ft3

74.7

cm

3

3.4.

357.

8 in

337

69.9

m3

5.6.

1231

.5 y

d312

10.6

mm

3

Fin

d t

he

volu

me

of e

ach

obl

iqu

e p

yram

id o

r co

ne.

Rou

nd

to

the

nea

rest

ten

th i

fn

eces

sary

.

7.8.

32 f

t345

2.4

cm3

12 c

m6 cm

4 ft

4 ft

6 ft

66!

18 m

m

25 y

d

14 y

d

25 m12

m

8 in

.10

in.

14 in

.

4 cm

7 cm

8 cm

5 ft

5 ft

8 ft

©G

lenc

oe/M

cGra

w-H

ill73

2G

lenc

oe G

eom

etry

Fin

d t

he

volu

me

of e

ach

pyr

amid

or

con

e.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

1.2.

343.

1 yd

323

95.8

cm

3

3.4.

1419

.4 f

t311

04.6

mm

3

5.6.

132

in3

4688

.3 f

t3

7.C

ON

STR

UC

TIO

NM

r.G

anty

bui

lt a

con

ical

sto

rage

she

d.T

he b

ase

of t

he s

hed

is 4

met

ers

in d

iam

eter

,and

the

hei

ght

of t

he s

hed

is 3

.8 m

eter

s.W

hat

is t

he v

olum

e of

the

she

d?ab

out

15.9

m3

8.H

ISTO

RYT

he s

tart

of

the

pyra

mid

age

beg

an w

ith

Kin

g Zo

ser’s

pyr

amid

,ere

cted

in t

he27

th c

entu

ry B

.C.I

n it

s or

igin

al s

tate

,it

stoo

d 62

met

ers

high

wit

h a

rect

angu

lar

base

that

mea

sure

d 14

0 m

eter

s by

118

met

ers.

Fin

d th

e vo

lum

e of

the

ori

gina

l pyr

amid

.ab

out

341,

413.

3 m

3

37 ft

11 ft

6 in

.6

in.

11 in

.

52!

12 m

m19

ft

9 ft

12.5

cm

25 c

m

23 c

m

9.2

yd9.

2 yd

13 y

d

Pra

ctic

e (A

vera

ge)

Volu

mes

of P

yram

ids

and

Con

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-2

13-2



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csVo

lum

es o

f Pyr

amid

s an

d C

ones

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-2

13-2

©G

lenc

oe/M

cGra

w-H

ill73

3G

lenc

oe G

eom

etry

Lesson 13-2

Pre-

Act

ivit

yH

ow d

o ar

chit

ects

use

geo

met

ry?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-2 a

t th

e to

p of

pag

e 69

6 in

you

r te

xtbo

ok.

In a

ddit

ion

to r

efle

ctin

g m

ore

light

,why

do

you

thin

k th

e ar

chit

ect

of t

heT

rans

amer

ica

Pyr

amid

may

hav

e de

sign

ed t

he b

uild

ing

as a

squ

are

pyra

mid

rath

er t

han

a re

ctan

gula

r pr

ism

?S

ampl

e an

swer

:The

pyr

amid

ism

ore

unus

ual a

nd h

as a

mor

e dr

amat

ic a

ppea

ranc

e,so

itat

trac

ts m

ore

atte

ntio

n.W

ith t

he s

harp

poi

nt a

t th

e to

p,it

seem

s to

soa

r up

into

the

sky

.

Rea

din

g t

he

Less

on

1.In

eac

h ca

se,t

wo

solid

s ar

e de

scri

bed.

Det

erm

ine

whe

ther

the

fir

st s

olid

or

the

seco

ndso

lid h

as t

he g

reat

er v

olum

e,or

if t

he t

wo

solid

s ha

ve t

he s

ame

volu

me.

(Ans

wer

by

wri

ting

firs

t,se

cond

,or

sam

e.)

a.F

irst

sol

id:A

rec

tang

ular

pri

sm w

ith

leng

th x

,wid

th y

,and

hei

ght

zSe

cond

sol

id:A

rec

tang

ular

pri

sm w

ith

leng

th 2

x,w

idth

y,h

eigh

t z

seco

ndb.

Fir

st s

olid

:a r

ecta

ngul

ar p

rism

tha

t ha

s a

squa

re b

ase

wit

h si

de le

ngth

xan

d th

atha

s he

ight

ySe

cond

sol

id:a

squ

are

pyra

mid

who

se b

ase

has

side

leng

th x

and

that

has

hei

ght

yfir

stc.

Fir

st s

olid

:a r

ight

con

e w

hose

bas

e ha

s ra

dius

xan

d th

at h

as h

eigh

t y

Seco

nd s

olid

:an

obliq

ue c

one

who

se b

ase

has

radi

us x

and

that

has

hei

ght

ysa

me

d.F

irst

sol

id:a

con

e w

hose

bas

e ha

s ra

dius

x,a

nd w

hose

hei

ght

is y

Seco

nd s

olid

:a c

ylin

der

who

se b

ases

hav

e ra

dius

x,a

nd w

hose

hei

ght

is y

seco

nde.

Fir

st s

olid

:a c

one

who

se b

ase

has

radi

us x

and

who

se h

eigh

t is

ySe

cond

sol

id:a

squ

are

pyra

mid

who

se b

ase

has

side

leng

th x

and

who

se h

eigh

t is

yfir

st

2.Su

pply

the

mis

sing

num

bers

to

form

tru

e st

atem

ents

.a.

If t

he le

ngth

,wid

th,a

nd h

eigh

t of

a r

ecta

ngul

ar b

ox a

re a

ll do

uble

d,it

s vo

lum

e w

ill

be m

ulti

plie

d by

.

b.If

the

rad

ius

of a

cyl

inde

r is

tri

pled

and

the

hei

ght

is u

ncha

nged

,the

vol

ume

will

be

mul

tipl

ied

by

.c.

In a

squ

are

pyra

mid

,if

the

side

leng

th o

f th

e ba

se is

mul

tipl

ied

by 1

.5 a

nd t

he h

eigh

t is

dou

bled

,the

vol

ume

will

be

mul

tipl

ied

by

.d.

In a

con

e,if

the

rad

ius

of t

he b

ase

is t

ripl

ed a

nd t

he h

eigh

t is

dou

bled

,the

vol

ume

will

be

mul

tipl

ied

by

.e.

In a

cub

e,if

the

edg

e le

ngth

is m

ulti

plie

d by

5,t

he v

olum

e w

ill b

e m

ulti

plie

d by

.

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stud

ents

fin

d it

eas

ier

to r

emem

ber

mat

hem

atic

al f

orm

ulas

if t

hey

can

put

them

in w

ords

.Use

wor

ds t

o de

scri

be in

one

sen

tenc

e ho

w t

o fi

nd t

he v

olum

e of

any

pyr

amid

or c

ylin

der.

Sam

ple

answ

er:M

ultip

ly t

he a

rea

of t

he b

ase

by t

he h

eigh

t an

ddi

vide

by

3.

125

18

4.5

9

8

©G

lenc

oe/M

cGra

w-H

ill73

4G

lenc

oe G

eom

etry

Frus

tum

sA

fru

stu

mis

a f

igur

e fo

rmed

whe

n a

plan

e in

ters

ects

a p

yram

id o

rco

ne s

o th

at t

he p

lane

is p

aral

lel t

o th

e so

lid’s

bas

e.T

he f

rust

um is

th

e pa

rt o

f th

e so

lid b

etw

een

the

plan

e an

d th

e ba

se.T

o fi

nd t

hevo

lum

e of

a f

rust

um,t

he a

reas

of

both

bas

es m

ust

be c

alcu

late

d an

dus

ed in

the

for

mul

a

V!

$1 3$ h(B

1#

B2

#"

B1B

2#

),w

here

h!

heig

ht (

perp

endi

cula

r di

stan

ce b

etw

een

the

base

s),

B1

!ar

ea o

f to

p ba

se,a

nd B

2!

area

of

bott

om b

ase.

Des

crib

e th

e sh

ape

of t

he

base

s of

eac

h f

rust

um

.Th

en f

ind

th

e vo

lum

e.R

oun

d t

o th

e n

eare

st t

enth

.

1.2.

rect

angl

es;6

17.5

cm

3

circ

les;

335.

8 in

3

3.4.

trap

ezoi

ds;1

51.6

m3

circ

les;

3480

.9 f

t3

12 ft

13 ft

7 ft

8 m

6 m

12 m

4.5

m2.

25 m3

m

5 m

3 in

.

7.5

in.

4.5

in.

13 c

m 6 cm

9 cm

5 cm

19.5

cm

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-2

13-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Volu

mes

of S

pher

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-3

13-3

©G

lenc

oe/M

cGra

w-H

ill73

5G

lenc

oe G

eom

etry

Lesson 13-3

Vo

lum

es o

f Sp

her

esA

sph

ere

has

one

basi

c m

easu

rem

ent,

the

leng

th o

f it

s ra

dius

.If

you

know

the

rad

ius

of a

sph

ere,

you

can

calc

ulat

e it

s vo

lum

e.

Volu

me

of

a S

pher

eIf

a sp

here

has

a v

olum

e of

Vcu

bic

units

and

a r

adiu

s of

run

its, t

hen

V!

$4 3$ "r3

.

Fin

d t

he

volu

me

of a

sp

her

e w

ith

rad

ius

8 ce

nti

met

ers.

V!

$4 3$ "r3

Vol

ume

of a

sph

ere

!$4 3$ "

(8)3

r!

8

!21

44.7

Sim

plify

.

The

vol

ume

is a

bout

214

4.7

cubi

c ce

ntim

eter

s.

A s

ph

ere

wit

h r

adiu

s 5

inch

es j

ust

fit

s in

sid

e a

cyli

nd

er.W

hat

is

the

dif

fere

nce

bet

wee

n t

he

volu

me

of t

he

cyli

nd

er a

nd

th

e vo

lum

e of

th

e sp

her

e? R

oun

d t

o th

e n

eare

st

cubi

c in

ch.

The

bas

e of

the

cyl

inde

r is

25"

in2

and

the

heig

ht is

10

in.,

so t

he

volu

me

of t

he c

ylin

der

is 2

50"

in3 .

The

vol

ume

of t

he s

pher

e is

$4 3$ "(5

)3

or $50

30" $in

3 .T

he d

iffe

renc

e in

the

vol

umes

is 2

50"

%$50

30" $or

abo

ut 2

62 in

3 .

Fin

d t

he

volu

me

of e

ach

sol

id.R

oun

d t

o th

e n

eare

st t

enth

.

1.2.

3.

523.

6 ft

345

2.4

in3

8578

.6 in

3

4.5.

6.

268.

1 cm

357

6.0

in3

243.

9 in

3

7.A

hem

isph

ere

wit

h ra

dius

16

cent

imet

ers

just

fit

s in

side

a r

ecta

ngul

ar p

rism

.Wha

t is

the

diff

eren

ce b

etw

een

the

volu

me

of t

he p

rism

and

the

vol

ume

of t

he h

emis

pher

e?R

ound

to

the

near

est

cubi

c ce

ntim

eter

.78

05 c

m3

8 in

.di

ffere

nce

betw

een

volu

me

of c

ube

and

volu

me

of s

pher

e

13 in

.5

in.

8 cm

16 in

.

6 in

.

5 ft

5 in

.

5 in

.5

in.

5 in

.

8 cmr

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill73

6G

lenc

oe G

eom

etry

Solv

e Pr

ob

lem

s In

volv

ing

Vo

lum

es o

f Sp

her

esIf

you

wan

t to

kno

w if

a s

pher

eca

n be

pac

ked

insi

de a

noth

er c

onta

iner

,or

if y

ou w

ant

to c

ompa

re t

he c

apac

ity

of a

sph

ere

and

anot

her

shap

e,yo

u ca

n co

mpa

re v

olum

es.

Com

par

e th

e vo

lum

es o

f th

e sp

her

e an

d

the

cyli

nd

er.D

eter

min

e w

hic

h q

uan

tity

is

grea

ter.

V!

$4 3$ "r3

Vol

ume

of s

pher

eV

!"

r2h

Vol

ume

of c

ylin

der

!"

r2(1

.5r)

h!

1.5r

!1.

5"r3

Sim

plify

.

Com

pare

$4 3$ "r3

wit

h 1.

5"r3

.Sin

ce $4 3$

is le

ss t

han

1.5,

it f

ollo

ws

that

th

e vo

lum

e of

the

sph

ere

is le

ss t

han

the

volu

me

of t

he c

ylin

der.

Com

par

e th

e vo

lum

e of

a s

ph

ere

wit

h r

adiu

s r

to t

he

volu

me

of e

ach

fig

ure

bel

ow.

Wh

ich

fig

ure

has

a g

reat

er v

olu

me?

1.2.

The

volu

me

of t

he h

emis

pher

e Th

e vo

lum

e of

the

sph

ere

is g

reat

er.

is g

reat

er.

3.4.

The

volu

me

of t

he s

pher

eTh

e vo

lum

e of

the

sph

ere

is

gre

ater

.is

gre

ater

.

5.6.

The

volu

me

of t

he c

ylin

der

cann

ot b

e de

term

ined

(If

a)

0.63

,is

gre

ater

.th

e vo

lum

e of

the

hem

isph

ere

isgr

eate

r.If

a*

0.63

,the

vol

ume

ofth

e sp

here

is g

reat

er.)

2ar

3r

r

r3r

r

rr

rr 2

2r

r1.

5r

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Volu

mes

of S

pher

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-3

13-3

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Volu

mes

of S

pher

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-3

13-3

©G

lenc

oe/M

cGra

w-H

ill73

7G

lenc

oe G

eom

etry

Lesson 13-3

Fin

d t

he

volu

me

of e

ach

sp

her

e or

hem

isp

her

e.R

oun

d t

o th

e n

eare

st t

enth

.

1.T

he r

adiu

s of

the

sph

ere

is 9

cen

tim

eter

s.30

53.6

cm

3

2.T

he d

iam

eter

of

the

sphe

re is

10

inch

es.

523.

6 in

3

3.T

he c

ircu

mfe

renc

e of

the

sph

ere

is 2

6 m

eter

s.29

6.8

m3

4.T

he r

adiu

s of

the

hem

isph

ere

is 7

fee

t.71

8.4

ft3

5.T

he d

iam

eter

of

the

hem

isph

ere

is 1

2 ki

lom

eter

s.45

2.4

km3

6.T

he c

ircu

mfe

renc

e of

the

hem

isph

ere

is 4

8 ya

rds.

933.

8 yd

3

7.8.

2226

.1 c

m3

446,

091.

2 ft

3

9.10

.

190.

9 in

378

1.7

m3

14.4

m

4.5

in.

94.8

ft16

.2 c

m

©G

lenc

oe/M

cGra

w-H

ill73

8G

lenc

oe G

eom

etry

Fin

d t

he

volu

me

of e

ach

sp

her

e or

hem

isp

her

e.R

oun

d t

o th

e n

eare

st t

enth

.

1.T

he r

adiu

s of

the

sph

ere

is 1

2.4

cent

imet

ers.

7986

.4 c

m3

2.T

he d

iam

eter

of

the

sphe

re is

17

feet

.25

72.4

ft3

3.T

he c

ircu

mfe

renc

e of

the

sph

ere

is 3

8 m

eter

s.92

6.6

m3

4.T

he d

iam

eter

of

the

hem

isph

ere

is 2

1 in

ches

.24

24.5

in3

5.T

he c

ircu

mfe

renc

e of

the

hem

isph

ere

is 1

8 m

illim

eter

s.49

.2 m

m3

6.7.

7832

.9 f

t332

94.8

cm

3

8.9.

8578

.6 m

367

1.3

mm

3

10.P

AC

KA

GIN

GA

mbe

r pl

ans

to s

hip

a m

ini-

bask

etba

ll sh

e bo

ught

for

her

nep

hew

.The

circ

umfe

renc

e of

the

bal

l is

24 in

ches

and

the

pac

kage

she

wan

ts t

o sh

ip it

in is

are

ctan

gula

r bo

x th

at m

easu

res

8 in

ches

&8

inch

es &

9 in

ches

.Will

the

bas

ketb

all f

it in

the

box?

Exp

lain

.Ye

s;th

e di

amet

er o

f th

e ba

ll is

abo

ut 7

.64

in.,

so t

he b

all w

ill f

it in

the

box

.

C "

43

mm

32 m

C "

58

cm12

.32

ft

Pra

ctic

e (A

vera

ge)

Volu

mes

of S

pher

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-3

13-3



Rea

din

g t

o L

earn

Math

emati

csVo

lum

es o

f Sph

eres

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-3

13-3

©G

lenc

oe/M

cGra

w-H

ill73

9G

lenc

oe G

eom

etry

Lesson 13-3

Pre-

Act

ivit

yH

ow c

an y

ou f

ind

th

e vo

lum

e of

Ear

th?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-3 a

t th

e to

p of

pag

e 70

2 in

you

r te

xtbo

ok.

How

wou

ld y

ou e

stim

ate

the

radi

us o

f E

arth

bas

ed o

n E

rato

sthe

nes’

esti

mat

e of

its

diam

eter

?S

ampl

e an

swer

:Use

a c

alcu

lato

r to

divi

de 4

6,25

0 km

by

2$.

Rea

din

g t

he

Less

on

1.N

ame

all s

olid

s fr

om t

he f

ollo

win

g lis

t fo

r w

hich

eac

h vo

lum

e fo

rmul

a ca

n be

use

d:pr

ism

,pyr

amid

,con

e,cy

lind

er,s

pher

e,he

mis

pher

e.

a.V

!B

hpr

ism

,cyl

inde

rb.

V!

$4 3$ "r3

sphe

re

c.V

!$1 3$ B

hpy

ram

id,c

one

d.V

!"

r2h

cylin

der

e.V

!$1 3$ "

r2h

cone

f.V

!$2 3$ "

r3he

mis

pher

e

2.L

et r

repr

esen

t th

e ra

dius

and

dre

pres

ent

the

diam

eter

of

a sp

here

.Det

erm

ine

whe

ther

each

for

mul

a be

low

can

be

used

to

find

the

vol

ume

of a

sph

ere,

a he

mis

pher

e,or

nei

ther

.

a.V

!$2"

3r3$

hem

isph

ere

b.V

!$1 6$ "

d3

sphe

re

c.V

!$1 3$ "

r3ne

ither

d.V

!$3 4$ "

r3ne

ither

e.V

!$" 1d 23 $

hem

isph

ere

f.V

!$4 3$ "

r2h

neith

er

3.C

ompa

re t

he v

olum

es o

f th

ese

thre

e so

lids.

The

n co

mpl

ete

the

sent

ence

bel

ow.

Of

the

thre

e so

lids

show

n ab

ove,

the

has

the

larg

est

volu

me

and

the

has

the

smal

lest

vol

ume.

Hel

pin

g Y

ou

Rem

emb

er

4.A

goo

d w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o so

meo

ne e

lse.

Supp

ose

that

you

r cl

assm

ate

Lor

etta

kno

ws

that

the

exp

ress

ions

$4 3$ "r3

and

4"r2

are

used

in f

indi

ng

mea

sure

men

ts r

elat

ed t

o sp

here

s,bu

t ca

n’t

rem

embe

r w

hich

one

is u

sed

to f

ind

the

surf

ace

area

of

a sp

here

and

whi

ch o

ne is

use

d to

fin

d th

e vo

lum

e.H

ow c

an y

ou h

elp

her

to r

emem

ber

whi

ch is

whi

ch?

Sam

ple

answ

er:L

ook

at t

he p

ower

s of

rin

the

two

expr

essi

ons.

The

expr

essi

on w

ith r

3w

ill g

ive

a m

easu

rem

ent

incu

bic

units

,so

it is

the

exp

ress

ion

for

volu

me.

The

expr

essi

on w

ith r

2

will

giv

e a

mea

sure

men

t in

squ

are

units

,so

it is

the

exp

ress

ion

for

surf

ace

area

.

cone

sphe

re

2r

r

rr

r

©G

lenc

oe/M

cGra

w-H

ill74

0G

lenc

oe G

eom

etry

Sph

eres

and

Den

sity

The

den

sity

of a

met

al is

a r

atio

of

its

mas

s to

its

volu

me.

For

exam

ple,

the

mas

s of

alu

min

um is

2.7

gra

ms

per

cubi

c ce

ntim

eter

.H

ere

is a

list

of

seve

ral m

etal

s an

d th

eir

dens

itie

s.

Alu

min

um2.

7 g/

cm3

Cop

per

8.96

g/c

m3

Gol

d19

.32

g/cm

3Ir

on7.

874

g/cm

3

Lea

d11

.35

g/cm

3P

lati

num

21.4

5 g/

cm3

Silv

er10

.50

g/cm

3

To c

alcu

late

the

mas

s of

a p

iece

of

met

al,m

ulti

ply

volu

me

by d

ensi

ty.

Fin

d t

he

mas

s of

a s

ilve

r ba

ll t

hat

is

0.8

cm

in d

iam

eter

.

M!

D'

V

!10

.5 '

$4 3$ "(0

.4)3

!10

.5 (

0.27

)!

2.83

The

mas

s is

abo

ut 2

.83

gram

s.

Fin

d t

he

mas

s of

eac

h m

etal

bal

l d

escr

ibed

.Ass

um

e th

e ba

lls

are

sph

eric

al.R

oun

d y

our

answ

ers

to t

he

nea

rest

ten

th.

1.a

copp

er b

all 1

.2 c

m in

dia

met

er8.

1 g

2.a

gold

bal

l 0.6

cm

in d

iam

eter

2.2

g

3.an

alu

min

um b

all w

ith

radi

us 3

cm

305.

4 g

4.a

plat

inum

bal

l wit

h ra

dius

0.7

cm

30.8

g

Sol

ve.A

ssu

me

the

ball

s ar

e sp

her

ical

.Rou

nd

you

r an

swer

s to

th

e n

eare

st t

enth

.

5.A

lead

bal

l wei

ghs

326

g.F

ind

the

radi

us o

f th

e ba

ll to

the

nea

rest

te

nth

of a

cen

tim

eter

.1.

9 cm

6.A

n ir

on b

all w

eigh

s 80

4 g.

Fin

d th

e di

amet

er o

f th

e ba

ll to

the

ne

ares

t te

nth

of a

cen

tim

eter

.5.

8 cm

7.A

silv

er b

all a

nd a

cop

per

ball

each

hav

e a

diam

eter

of

3.5

cm.

Whi

ch w

eigh

s m

ore?

How

muc

h m

ore?

silv

er;3

4.6

g

8.A

n al

umin

um b

all a

nd a

lead

bal

l eac

h ha

ve a

rad

ius

of 1

.2 c

m.

Whi

ch w

eigh

s m

ore?

How

muc

h m

ore?

lead

;62.

6g

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-3

13-3

Exam

ple

Exam

ple



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Con

grue

nt a

nd S

imila

r S

olid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-4

13-4

©G

lenc

oe/M

cGra

w-H

ill74

1G

lenc

oe G

eom

etry

Lesson 13-4

Co

ng

ruen

t o

r Si

mila

r So

lids

If t

he c

orre

spon

ding

ang

les

and

side

s of

tw

o so

lids

are

cong

ruen

t,th

en t

he s

olid

s ar

e co

ngru

ent.

Als

o,th

e co

rres

pond

ing

face

s ar

e co

ngru

ent

and

thei

r su

rfac

e ar

eas

and

volu

mes

are

equ

al.S

olid

s th

at h

ave

the

sam

e sh

ape

but

are

diff

eren

t si

zes

are

sim

ilar

.You

can

det

erm

ine

whe

ther

tw

o so

lids

are

sim

ilar

by c

ompa

ring

the

rati

o,or

sca

le f

acto

r,of

cor

resp

ondi

ng li

near

mea

sure

men

ts.

Des

crib

e ea

ch p

air

of s

olid

s.

•F

igur

es I

and

II

are

sim

ilar

beca

use

the

figu

res

have

the

sam

e sh

ape.

The

rat

io o

f ea

chpa

ir o

f co

rres

pond

ing

side

s is

1:3

.•

Fig

ures

III

and

IV

are

con

grue

nt b

ecau

se t

hey

have

the

sam

e sh

ape

and

all c

orre

spon

ding

mea

sure

men

ts a

re t

he s

ame.

•F

igur

es V

and

VI

are

not

cong

ruen

t,an

d th

ey a

re n

ot s

imila

r be

caus

e $4 8$

($1 12 2$

.

Det

erm

ine

wh

eth

er e

ach

pai

r of

sol

ids

are

sim

ila

r,co

ngr

uen

t,or

nei

ther

.

1.2.

sim

ilar

neith

er

3.4.

cong

ruen

tco

ngru

ent

5.6.

neith

ersi

mila

r

2 7

21

6

5

8

5

8

4

4

88

5

5

2

2

2 26

6

7

7

12

4

5

1

106

8

53

4

III

IIIIV

VV

Isi

mila

rco

ngru

ent

non-

sim

ilar

125

5

512

124

85

5

5

7

7

9

6

43

2

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill74

2G

lenc

oe G

eom

etry

Pro

per

ties

of

Sim

ilar

Solid

sT

hese

tw

o so

lids

are

sim

ilar

wit

h a

scal

e fa

ctor

of

1:2.

The

sur

face

ar

eas

are

62 c

m2

and

248

cm2

and

the

volu

mes

are

30

cm

3an

d 24

0 cm

3 .N

otic

e th

at t

he r

atio

of

the

surf

ace

area

s is

62

:248

,whi

ch is

1:4

or

12:2

2 ,an

d th

e ra

tio

of t

he v

olum

es is

30

:240

,whi

ch is

1:8

or

13:2

3 .

If tw

o so

lids

are

sim

ilar

with

a s

cale

fact

or o

f a:b

, the

n th

e su

rfac

e ar

eas

have

a r

atio

of a

2:b

2 , a

nd th

e vo

lum

es h

ave

a ra

tio o

f a3:b

3 .

Use

th

e tw

o sp

her

es.

a.F

ind

th

e sc

ale

fact

or f

or t

he

two

sph

eres

.T

he s

cale

fac

tor

for

the

two

sphe

res

is t

he s

ame

as

the

rati

o of

the

ir r

adii,

or 5

:3.

b.F

ind

th

e ra

tio

of t

he

surf

ace

area

s of

th

e tw

o sp

her

es.

The

rat

io o

f th

e su

rfac

e ar

eas

is 5

2 :32

or 2

5:9.

c.F

ind

th

e ra

tio

of t

he

volu

mes

of

the

two

sph

eres

.T

he r

atio

of

the

volu

mes

is 5

3 :33

or 1

25:2

7.

Fin

d t

he

scal

e fa

ctor

for

eac

h p

air

of s

imil

ar f

igu

res.

Th

en f

ind

th

e ra

tio

of t

hei

rsu

rfac

e ar

eas

and

th

e ra

tio

of t

hei

r vo

lum

es.

1.2.

3:4;

9:16

;27:

647:

4;49

:16;

343:

64

3.4.

4:5;

16:2

5;64

:125

2:1;

4:1;

8:1

5.6.

5:4;

25:1

6;12

5:64

1:2:

1:4;

1:88

65

3

1215

4 yd

16 y

d

15 m

12 m

7 in

.4

in.

3 ft

4 ft

5 cm

3 cm

10 c

m5

cm2

cm3 cm

6 cm4

cm

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Con

grue

nt a

nd S

imila

r S

olid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-4

13-4

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Con

grue

nt a

nd S

imila

r S

olid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-4

13-4

©G

lenc

oe/M

cGra

w-H

ill74

3G

lenc

oe G

eom

etry

Lesson 13-4

Det

erm

ine

wh

eth

er e

ach

pai

r of

sol

ids

are

sim

ila

r,co

ngr

uen

t,or

nei

ther

.

1.si

mila

r

2.ne

ither

3.si

mila

r

4.co

ngru

ent

For

Exe

rcis

es 5

–8,r

efer

to

the

two

sim

ilar

pri

sms.

5.F

ind

the

scal

e fa

ctor

of

the

two

pris

ms.

#3 2#

6.F

ind

the

rati

o of

the

sur

face

are

as.

#9 4#

7.F

ind

the

rati

o of

the

vol

umes

.

#2 87 #

8.Su

ppos

e th

e vo

lum

e of

the

larg

er p

rism

is 8

10 c

ubic

cen

tim

eter

s.F

ind

the

volu

me

of t

hesm

alle

r pr

ism

.

240

cm3

15 c

m12

cm

9 cm

10 c

m8

cm6

cm

18 in

.

16 in

.16

in.

9 in

.

6 m

m

6 m

m

4 m

m9

mm

12 ft

12 ft

14 ft

20 ft

20 ft

21 ft

20 c

m20

cm

10 c

m

4 cm

40 c

m

8 cm

©G

lenc

oe/M

cGra

w-H

ill74

4G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er e

ach

pai

r of

sol

ids

are

sim

ila

r,co

ngr

uen

t,or

nei

ther

.

1.co

ngru

ent

2.si

mila

r

3.ne

ither

4.si

mila

r

For

Exe

rcis

es 5

–8,r

efer

to

the

two

sim

ilar

pri

sms.

5.F

ind

the

scal

e fa

ctor

of

the

two

pris

ms.

#5 3#

6.F

ind

the

rati

o of

the

sur

face

are

as.

#2 95 #

7.F

ind

the

rati

o of

the

vol

umes

.#1 22 75 #

8.Su

ppos

e th

e su

rfac

e ar

ea o

f th

e la

rger

pri

sm is

256

0 sq

uare

met

ers.

Fin

d th

e su

rfac

ear

ea o

f th

e sm

alle

r pr

ism

.92

1.6

m2

9.M

INIA

TUR

ESF

rank

Llo

yd W

righ

t de

sign

ed e

very

asp

ect

of t

he I

mpe

rial

Hot

el in

Tok

yo,

incl

udin

g th

e ch

airs

.The

dim

ensi

ons

of a

min

iatu

re I

mpe

rial

Hot

el c

hair

are

6.2

5 in

ches

&3

inch

es &

2.5

inch

es.I

f th

e sc

ale

of t

he r

eplic

a is

1:6

,wha

t ar

e th

e di

men

sion

s of

the

orig

inal

cha

ir?

37.5

in.%

18 in

.%15

in.

20 m

20 m

22 m

12 m

12 m

13.2

m

7.5

cm

20 c

m

15 c

m

4.5

cm

12 c

m9

cm

18 ft

24 ft

24 ft

9 ft

12 m

12 m

15 m

2.5

m

2 m

9.6

m

25 in

.15 in

.

30 in

.20

in.

Pra

ctic

e (A

vera

ge)

Con

grue

nt a

nd S

imila

r S

olid

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-4

13-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csC

ongr

uent

and

Sim

ilar

Sol

ids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-4

13-4

©G

lenc

oe/M

cGra

w-H

ill74

5G

lenc

oe G

eom

etry

Lesson 13-4

Pre-

Act

ivit

yH

ow a

re s

imil

ar s

olid

s ap

pli

ed t

o m

inia

ture

col

lect

ible

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-4 a

t th

e to

p of

pag

e 70

7 in

you

r te

xtbo

ok.

If y

ou w

ant

to m

ake

a m

inia

ture

wit

h a

scal

e fa

ctor

of

1:64

,how

can

you

use

the

actu

al o

bjec

t to

fin

d th

e m

easu

rem

ents

you

sho

uld

use

to c

onst

ruct

the

min

iatu

re?

Sam

ple

answ

er:T

ake

linea

r m

easu

rem

ents

of

the

actu

al o

bjec

t.D

ivid

e ea

ch m

easu

rem

ent

by 6

4 to

fin

d th

eco

rres

pond

ing

mea

sure

men

t fo

r th

e m

inia

ture

.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

heth

er e

ach

stat

emen

t is

alw

ays,

som

etim

es,o

r ne

ver

true

.a.

Tw

o cu

bes

are

sim

ilar.

alw

ays

b.T

wo

cone

s ar

e si

mila

r.so

met

imes

c.T

wo

cylin

ders

in w

hich

the

hei

ght

is t

wic

e th

e di

amet

er a

re s

imila

r.al

way

sd.

Tw

o cy

linde

rs w

ith

the

sam

e vo

lum

e ar

e co

ngru

ent.

som

etim

ese.

A p

rism

wit

h a

squa

re b

ase

and

a sq

uare

pyr

amid

are

sim

ilar.

neve

rf.

Tw

o re

ctan

gula

r pr

ism

s w

ith

equa

l sur

face

are

as a

re s

imila

r.so

met

imes

g.N

onsi

mila

r so

lids

have

dif

fere

nt v

olum

es.

som

etim

esh

.T

wo

hem

isph

eres

wit

h th

e sa

me

radi

us a

re c

ongr

uent

.al

way

s

2.Su

pply

the

mis

sing

rat

ios.

a.If

the

rat

io o

f th

e di

amet

ers

of t

wo

sphe

res

is 3

:1,t

hen

the

rati

o of

the

ir s

urfa

ce a

reas

is

,and

the

rat

io o

f th

eir

volu

mes

is

.

b.If

the

rat

io o

f th

e ra

dii o

f tw

o he

mis

pher

es is

2:5

,the

n th

e ra

tio

of t

heir

sur

face

are

as

is

,and

the

rat

io o

f th

eir

volu

mes

is

.

c.If

tw

o co

nes

are

sim

ilar

and

the

rati

o of

the

ir h

eigh

ts is

$4 3$ ,th

en t

he r

atio

of

thei

r

volu

mes

is

,and

the

rat

io o

f th

eir

surf

ace

area

s is

.

d.If

tw

o cy

linde

rs a

re s

imila

r an

d th

e ra

tio

of t

heir

sur

face

are

as is

100

:49,

then

the

rati

o of

the

rad

ii of

the

ir b

ases

is

,and

the

rat

io o

f th

eir

volu

mes

is

.

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al c

once

pt is

to

rela

te it

to

som

ethi

ng y

oual

read

y kn

ow.H

ow c

an w

hat

you

know

abo

ut t

he u

nits

use

d to

mea

sure

leng

ths,

area

s,an

d vo

lum

es h

elp

you

to r

emem

ber

the

theo

rem

abo

ut t

he r

atio

s of

sur

face

are

as a

ndvo

lum

es o

f si

mila

r so

lids?

Sam

ple

answ

er:L

engt

hs a

re m

easu

red

in li

near

units

,sur

face

are

as in

squ

are

units

,and

vol

umes

in c

ubic

uni

ts.T

ake

the

scal

e fa

ctor

,whi

ch is

the

rat

io o

f lin

ear

mea

sure

men

ts in

the

sol

ids,

and

squa

re it

to

get

the

ratio

of

thei

r su

rfac

e ar

eas

or c

ube

it to

get

the

rat

ioof

the

ir v

olum

es.

1000

:343

10:7

#1 96 ##6 24 7#

8:12

54:

25

27:1

9:1

©G

lenc

oe/M

cGra

w-H

ill74

6G

lenc

oe G

eom

etry

Con

grue

nt a

nd S

imila

r S

olid

s

Det

erm

ine

wh

eth

er e

ach

pai

r of

sol

ids

is s

imil

ar,

con

gru

ent,

or n

eith

er.

1.2.

neith

ersi

mila

r

3.4.

cong

ruen

tsi

mila

r

Th

e tw

o re

ctan

gula

r p

rism

s sh

own

at

the

righ

t ar

e si

mil

ar.

5.F

ind

the

rati

o of

the

per

imet

ers

of t

he b

ases

.7:

5

6.W

hat

is t

he r

atio

of

the

surf

ace

area

s?72

:52

or 4

9:25

7.Su

ppos

e th

e vo

lum

e of

the

sm

alle

r pr

ism

is 6

0 in

3 .F

ind

the

volu

me

of t

he la

rger

pri

sm.

164.

64 in

3

Det

erm

ine

wh

eth

er e

ach

sta

tem

ent

is t

rue

or f

als

e.If

th

e st

atem

ent

is f

alse

,rew

rite

it

so t

hat

it

is t

rue.

8.If

tw

o cy

linde

rs a

re s

imila

r,th

en t

heir

vol

umes

are

equ

al.

Fals

e;if

two

cylin

ders

are

con

grue

nt,t

hen

thei

r vo

lum

es a

re e

qual

.

9.D

oubl

ing

the

heig

ht o

f a

cylin

der

doub

les

the

volu

me.

true

10.T

wo

solid

s ar

e co

ngru

ent

if t

hey

have

the

sam

e sh

ape.

Fals

e;tw

o so

lids

are

sim

ilar

if th

ey h

ave

the

sam

e sh

ape.

7 in

.5

in.

24 y

d

12 y

d

12 y

d6

yd 8 yd

16 y

d12

m

3 m

3 m

3 m

3 m

3 m

3 m

4 m

4 m

4 m

4 m

4 m

10 m

48 m

16 m15

m

14 c

m

11 c

m

7 cm

7 cm

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-4

13-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Coo

rdin

ates

in S

pace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-5

13-5

©G

lenc

oe/M

cGra

w-H

ill74

7G

lenc

oe G

eom

etry

Lesson 13-5

Gra

ph

So

lids

in S

pac

eIn

spa

ce,y

ou c

an d

escr

ibe

the

loca

tion

of

a po

int

usin

g an

ord

ered

tri

ple

of r

eal

num

bers

.The

x-,

y-,a

nd z

-axe

s ar

e pe

rpen

dicu

lar

to

each

oth

er,a

nd t

he c

oord

inat

es f

or p

oint

Par

e th

e or

dere

d tr

iple

(%

4,6,

5).A

rec

tang

ular

pri

sm c

an b

edr

awn

to s

how

per

spec

tive

.

Gra

ph

th

e re

ctan

gula

r so

lid

th

at

con

tain

s th

e or

der

ed t

rip

le (

2,1,

%2)

an

d t

he

orig

in.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.•

Plo

t th

e x-

coor

dina

te f

irst

.Dra

w a

sol

id s

egm

ent

from

the

ori

gin

2 un

its

in t

he p

osit

ive

dire

ctio

n.•

Plo

t th

e y-

coor

dina

te n

ext.

Dra

w a

sol

id s

egm

ent

1 un

it in

the

pos

itiv

e di

rect

ion.

•P

lot

the

z-co

ordi

nate

nex

t.D

raw

a s

olid

seg

men

t 2

unit

s in

the

neg

ativ

e di

rect

ion.

•D

raw

the

rec

tang

ular

pri

sm,u

sing

dot

ted

lines

for

hi

dden

edg

es o

f th

e pr

ism

.•

Lab

el t

he c

oord

inat

es o

f ea

ch v

erte

x.

Gra

ph

th

e re

ctan

gula

r so

lid

th

at c

onta

ins

the

give

n p

oin

t an

d t

he

orig

in a

sve

rtic

es.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.

1.A

(2,1

,3)

2.G

(%1,

2,3)

3.P

(%2,

1,%

1)4.

T(%

1,3,

2)

y

x

z

( 0, 0

, 0)

( 0, 3

, 0)

( &1,

3, 0

)

( 0, 0

, 2)

( 0, 3

, 2)

( &1,

0, 2

)

( &1,

0, 0

)

T( &

1, 3

, 2)

y

x

z

( 0, 0

, 0)

( 0, 1

, 0)

P( &

2, 1

, &1)

( &2,

0, &

1)( &

2, 1

, 0)

( &2,

0, 0

)

( 0, 0

, &1)

( 0, 1

, &1)

y

x

z

( 0, 0

, 0)

( 0, 0

, 3)

( &1,

2, 0

)( &

1, 0

, 0)

( &1,

0, 3

)G

( &1,

2, 3

)

( 0, 2

, 0)

( 0, 2

, 3)

y

x

z( 0

, 0, 3

)

( 0, 0

, 0)

( 2, 0

, 3)

( 2, 0

, 0)

( 2, 1

, 0)

( 0, 1

, 0)

( 0, 1

, 3)

A( 2

, 1, 3

)

y

x

z

( 0, 0

, 0)

( 0, 1

, 0)

( 0, 1

, &2)

( 2, 1

, &2)

( 2, 0

, &2)

( 0, 0

, &2)

( 2, 0

, 0)

( 2, 1

, 0)

y

x

z O

P( &

4, 6

, 5)

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill74

8G

lenc

oe G

eom

etry

Dis

tan

ce a

nd

Mid

po

int

Form

ula

sYo

u ca

n ex

tend

the

Dis

tanc

e Fo

rmul

a an

d th

eM

idpo

int

Form

ula

to t

hree

dim

ensi

ons

to f

ind

the

dist

ance

bet

wee

n tw

o po

ints

in s

pace

an

d to

fin

d th

e m

idpo

int

of t

he s

egm

ent

conn

ecti

ng t

wo

poin

ts.

Dis

tanc

e Fo

rmul

aG

iven

two

poin

ts A

(x1,

y1,

z1)

and

B(x

2, y

2, z

2) in

spa

ce, t

he d

ista

nce

betw

een

in S

pace

Aan

d B

is g

iven

by

AB

!"

(x1

%#

x 2)2

##

(y1

%#

y 2)2

##

(z1

%#

z 2)2

#.

Mid

poin

t Fo

rmul

aG

iven

two

poin

ts A

(x1,

y1,

z1)

and

B(x

2, y

2, z

2) in

spa

ce, t

he m

idpo

int o

f A#B#

is

in S

pace

at $$x 1

# 2x 2

$, $

y 1# 2

y 2$

, $z 1

# 2z 2

$%.

Det

erm

ine

the

dis

tan

ce b

etw

een

A(3

,2,!

5) a

nd

B(%

4,6,

9).

Th

en d

eter

min

e th

e co

ord

inat

es o

f th

e m

idp

oin

t of

A #B#

.

AB

!"

(x1

%#

x 2)2

##

(y1

%#

y 2)2

##

(z1

%#

z 2)2

#!

"(3

%(

#%

4))2

##

(2 %

#6)

2#

#(%

5 %

#9)

2#

!"

72#

(#

%4)

2#

#(%

14#

)2 #!

"49

##

16 #

#19

6#

!16

.2

mid

poin

t of

A#B#

!$$x 1

# 2x 2

$,$

y 1# 2

y 2$

,$z 1

# 2z 2

$%

!$$3

#2(%

4)$

,$2

# 26

$,$

%5 2#

9$

%!

(%0.

5,4,

2)

Det

erm

ine

the

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts.T

hen

det

erm

ine

the

coor

din

ates

of

the

mid

poi

nt

Mof

th

e se

gmen

t jo

inin

g th

e p

air

of p

oin

ts.

1.A

(0,7

,%4)

and

B(%

2,8,

3)2.

C(%

7,6,

5) a

nd D

(10,

2,%

5)

AB

!#

54$!

7.3

;M!%

1,#1 25 #

,%#1 2# "

CD

!#

405

$!

20.1

;M!#3 2# ,

4,0 "

3.E

(3,1

,%2)

and

F(%

2,3,

4)4.

G(%

4,1,

1) a

nd H

(0,2

,%1)

EF

!#

65$!

8.1

;M!#1 2# ,

2,1 "

GH

!#

21$!

4.6;

M!%

2,#3 2# ,

0 "

5.J(

6,1,

%2)

and

K(%

1,%

2,1)

6.L

(%5,

0,%

3) a

nd N

(0,0

,%4)

JK!

#67$

!8.

2 ;M

!#5 2# ,%

#1 2# ,%

#1 2# "LN

!#

26$!

5.1;

M!%

#5 2# ,0,

%#7 2# "

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Coo

rdin

ates

in S

pace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-5

13-5

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Coo

rdin

ates

in S

pace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-5

13-5

©G

lenc

oe/M

cGra

w-H

ill74

9G

lenc

oe G

eom

etry

Lesson 13-5

Gra

ph

th

e re

ctan

gula

r so

lid

th

at c

onta

ins

the

give

n p

oin

t an

d t

he

orig

in a

sve

rtic

es.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.

1.A

(%5,

3,2)

2.H

(3,2

,5)

3.D

ilate

the

pri

sm b

y a

scal

e fa

ctor

of

2.G

raph

the

imag

e un

der

the

dila

tion

.

A'(

&4,

6,2)

,B'(

&4,

0,2)

,C'(

0,0,

2),D

'(0,

6,2)

,E

'(&

4,6,

0),F

'(&

4,0,

0),G

'(0,

0,0)

,H'(

0,6,

0)

Det

erm

ine

the

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts.T

hen

det

erm

ine

the

coor

din

ates

of

the

mid

poi

nt

Mof

th

e se

gmen

t jo

inin

g th

e p

air

of p

oin

ts.

4.R

(2,1

,0)

and

S(3

,3,4

)5.

Q(5

,0,%

2) a

nd T

(2,3

,2)

RS

"#

21$; !#5 2# ,

2,2 "

QT

"#

34$; !#7 2# ,

#3 2# ,0 "

6.A

(%4,

1,6)

and

B(%

1,0,

4)7.

J(0,

5,1)

and

K(4

,%3,

2)

AB

"#

14$; !&

#5 2# ,#1 2# ,

5 "JK

"9;

!2,1,

#3 2# "

y

x

z

A'

B'

C'

D'

E'

F'

G'

H'

y

x

z

AB

C

GH

ED

F

y

x

z

N( 0

, 0, 0

)

M( 3

, 0, 0

)L(

3, 2

, 0)

P( 0

, 2, 0

)

K( 0

, 2, 5

)J(

0, 0

, 5)

I(3,

0, 5

)

H( 3

, 2, 5

)

y

x

zA

( &5,

3, 2

)

E( &

5, 3

, 0)

B( &

5, 0

, 2)

C( 0

, 0, 2

)D

( 0, 3

, 2)

H( 0

, 3, 0

)

G( 0

, 0, 0

)

F( &

5, 0

, 0)

©G

lenc

oe/M

cGra

w-H

ill75

0G

lenc

oe G

eom

etry

Gra

ph

th

e re

ctan

gula

r so

lid

th

at c

onta

ins

the

give

n p

oin

t an

d t

he

orig

in a

sve

rtic

es.L

abel

th

e co

ord

inat

es o

f ea

ch v

erte

x.

1.E

(4,6

,%2)

2.R

(%3,

%5,

4)

Det

erm

ine

the

dis

tan

ce b

etw

een

eac

h p

air

of p

oin

ts.T

hen

det

erm

ine

the

coor

din

ates

of

the

mid

poi

nt

Mof

th

e se

gmen

t jo

inin

g th

e p

air

of p

oin

ts.

3.Y

(%5,

1,2)

and

Z(3

,%3,

1)4.

E(4

,2,0

) an

d F

(3,2

,%2)

YZ

"9;

!&1,

&1,

#3 2# "E

F"

#5$;

!#7 2# ,2,

&1 "

5.B

(%2,

%2,

%3)

and

C(1

,%3,

0)6.

H(2

,0,%

3) a

nd I

(4,%

1,5)

BC

"#

19$; !&

#1 2# ,&

#5 2# ,&

#3 2# "H

I"#

69$; !3

,&#1 2# ,

1 "

7.A

NIM

ATI

ON

Der

ek w

ants

to

anim

ate

an im

age

for

his

scie

nce

pres

enta

tion

by

mov

ing

it f

rom

one

pos

itio

n to

ano

ther

.The

mes

h of

the

imag

e is

a r

ecta

ngul

ar p

rism

wit

hco

ordi

nate

s A

(%3,

2,3)

,B(%

3,0,

3),C

(0,0

,3),

D(0

,2,3

),E

(%3,

2,0)

,F(%

3,0,

0),G

(0,0

,0),

and

H(0

,2,0

).F

ind

the

coor

dina

tes

of t

he m

esh

afte

r th

e tr

ansl

atio

n (x

,y,z

) →(x

%7,

y,z)

.G

raph

bot

h th

e pr

eim

age

and

imag

e of

the

mes

h.A

'(&

10,2

,3),

B'(

&10

,0,3

),C

'(&

7,0,

3),

D'(

&7,

2,3)

,E'(

&10

,2,0

),F

'(&

10,0

,0),

G'(

&7,

0,0)

,H'(

&7,

2,0)

y

x

zA

'B

'

C'

D'

E'

F'

G'

H'

AB

CD

EF

GH

y

x

z

X( 0

, 0, 0

)

Y( 0

, &5,

0)

V( &

3, &

5, 0

)R( &

3, &

5, 4

)S

( &3,

0, 4

)

T( 0

, 0, 4

)U

( 0, &

5, 4

)

W( &

3, 0

, 0)

y

x

z

K( 0

, 0, 0

)

J(4,

0, 0

)I(

4, 6

, 0)

L(0,

6, 0

)

H( 0

, 6, &

2)

E( 4

, 6, &

2)F

( 4, 0

, &2)

G( 0

, 0, &

2)

Pra

ctic

e (A

vera

ge)

Coo

rdin

ates

in S

pace

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-5

13-5



Rea

din

g t

o L

earn

Math

emati

csC

oord

inat

es in

Spa

ce

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-5

13-5

©G

lenc

oe/M

cGra

w-H

ill75

1G

lenc

oe G

eom

etry

Lesson 13-5

Pre-

Act

ivit

yH

ow i

s th

ree-

dim

ensi

onal

gra

ph

ing

use

d i

n c

omp

ute

r an

imat

ion

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 13

-5 a

t th

e to

p of

pag

e 71

4 in

you

r te

xtbo

ok.

Why

wou

ld a

mes

h be

cre

ated

fir

st?

Sam

ple

answ

er:A

mes

h is

an

outli

ne t

hat

the

anim

ator

wou

ld u

se f

irst

like

a s

ketc

h be

fore

rend

erin

g a

final

imag

e.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.M

atch

eac

h po

int

from

the

fir

st c

olum

n w

ith

its

coor

dina

tes

from

the

sec

ond

colu

mn.

a.A

iiii.

(3,0

,0)

b.B

viii

.(3,

0,%

4)

c.O

vii

i.(3

,%2,

0)

d.J

iiv

.(3

,%2,

%4)

e.H

ivv.

(0,0

,0)

f.K

iivi

.(0,

%2,

0)

g.T

viii

vii.

(0,%

2,%

4)

h.

Rvi

ivi

ii.(

0,0,

%4)

2.W

hich

of

the

follo

win

g ex

pres

sion

s gi

ve t

he d

ista

nce

betw

een

the

poin

ts a

t (4

,%1,

%5)

and

(%3,

2,%

9)?

A,E

,H

A."

72#

(#

%3)

2#

#42 #

B."

12#

1#

2#

(%#

14)2

#

C."

22#

2#

2#

42#

D. $$1 2$ ,

$1 2$ ,%

7 %E

."(%

3 %

#4)

2#

#(%

1 %

#2)

2#

#(%

9 #

#5)

2#

F."

24#

G."

(%3

##

4)2

##

[2 #

#(%

1)]2

##

[%#

9 #

(%#

5)]2

#H

."74#

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

new

mat

hem

atic

al f

orm

ulas

is t

o re

late

the

m t

o on

es y

oual

read

y kn

ow.H

ow c

an y

ou u

se y

our

know

ledg

e of

the

Dis

tanc

e an

d M

idpo

int

Form

ulas

in t

wo

dim

ensi

ons

to r

emem

ber

the

form

ulas

in t

hree

dim

ensi

ons?

Sam

ple

answ

er:

Sta

rt w

ith t

he fo

rmul

as fo

r tw

o di

men

sion

s.A

dd a

thi

rd t

erm

und

er t

hera

dica

l in

the

Dis

tanc

e Fo

rmul

a an

d a

thir

d co

ordi

nate

in t

he M

idpo

int

Form

ula

that

is ju

st li

ke t

he o

ther

tw

o ex

cept

tha

t th

e va

riab

le is

zra

ther

than

xor

y.

y

x

z

A

KBO

R

H

T

J

©G

lenc

oe/M

cGra

w-H

ill75

2G

lenc

oe G

eom

etry

Pla

nes

and

Cyl

indr

ical

Sur

face

sC

onsi

der

the

poin

ts (x

,y,z

) in

spa

ce w

hose

co

ordi

nate

s sa

tisf

y th

e eq

uati

on z

!1.

Sinc

e x

and

ydo

not

occ

ur in

the

equ

atio

n,an

y po

int

wit

h it

s z-

coor

dina

te e

qual

to

1 ha

s co

ordi

nate

s th

at s

atis

fy t

he e

quat

ion.

The

se a

re t

he p

oint

s in

the

pla

ne 1

uni

t ab

ove

the

xy-p

lane

.Thi

s pl

ane

is p

erpe

ndic

ular

to

the

z-ax

is a

t (0

,0,1

).

Nex

t co

nsid

er t

he p

oint

s (x

,y,z

) w

hose

co

ordi

nate

s sa

tisf

y x2

#y2

!16

.In

the

xy-p

lane

,al

l poi

nts

on t

he c

ircl

e w

ith

cent

er (

0,0,

0) a

ndra

dius

4 h

ave

coor

dina

tes

that

sat

isfy

the

eq

uati

on.I

n th

e pl

ane

perp

endi

cula

r to

the

z-

axis

at

(0,0

,k),

the

poin

ts t

hat

sati

sfy

the

equa

tion

are

tho

se o

n th

e ci

rcle

wit

h ce

nter

(0

,0,k

) an

d ra

dius

4.T

he g

raph

in s

pace

of

x2#

y2!

16 is

an

infi

nite

cyl

indr

ical

sur

face

w

hose

axi

s is

the

z-a

xis

and

who

se r

adiu

s is

4.

Des

crib

e th

e gr

aph

in

sp

ace

of e

ach

equ

atio

n.Y

ou m

ay f

ind

it

hel

pfu

l to

mak

e sk

etch

es o

n a

sep

arat

e sh

eet.

1.x

!5

the

plan

e pe

rpen

dicu

lar

to t

he x

-axi

s at

(5,

0,0)

2.y

!%

2th

e pl

ane

perp

endi

cula

r to

the

y-a

xis

at (

0,&

2,0)

3.x

#y

!7

the

plan

e pa

ralle

l to

the

z-ax

is a

nd c

onta

inin

g th

e lin

e th

roug

h(0

,7,0

) an

d (7

,0,0

)

4.z2

#y2

!25

the

infin

ite c

ylin

dric

al s

urfa

ce w

hose

axi

s is

the

x-a

xis

and

who

se r

adiu

s is

5

5.(x

%2)

2#

(y%

5)2

!1

the

infin

ite c

ylin

dric

alsu

rfac

e w

hose

axi

s is

the

line

para

llel t

o th

e z-

axis

and

pas

sing

thr

ough

(2,

5,0)

and

who

se r

adiu

s is

1

6.x2

#y2

#z2

!0

the

poin

t (0

,0,0

)

z

y

x

O

(0, 0

, k)

plan

e fo

r z "

k

z

y

x

O

(0, 0

, 1)

N

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

13-5

13-5


Documents

Chapter 13 Resource Masters - Math Class©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the Chapter 13 Resource Masters The Fast File Chapter Resource system allows