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
Lesson 13-2Study Guide and Intervention . . . . . . . . 729–730Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 731Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 732Reading to Learn Mathematics . . . . . . . . . . 733Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 734
Lesson 13-3Study Guide and Intervention . . . . . . . . 735–736Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 737Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 738Reading to Learn Mathematics . . . . . . . . . . 739Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 740
Lesson 13-4Study Guide and Intervention . . . . . . . . 741–742Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 743Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 744Reading to Learn Mathematics . . . . . . . . . . 745Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 746
Lesson 13-5Study Guide and Intervention . . . . . . . . 747–748Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 749Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 750Reading to Learn Mathematics . . . . . . . . . . 751Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 752
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
© Glencoe/McGraw-Hill 724 Glencoe Geometry
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
© Glencoe/McGraw-Hill 725 Glencoe Geometry
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
© Glencoe/McGraw-Hill 726 Glencoe Geometry
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
© Glencoe/McGraw-Hill 727 Glencoe Geometry
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?
© Glencoe/McGraw-Hill 728 Glencoe Geometry
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
© Glencoe/McGraw-Hill 729 Glencoe Geometry
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
© Glencoe/McGraw-Hill 730 Glencoe Geometry
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
Study Guide and Intervention (continued)
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
© Glencoe/McGraw-Hill 731 Glencoe Geometry
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
© Glencoe/McGraw-Hill 732 Glencoe Geometry
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
© Glencoe/McGraw-Hill 733 Glencoe Geometry
Less
on
13-
2
Pre-Activity How do architects use geometry?
Read the introduction to Lesson 13-2 at the top of page 696 in your textbook.
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.
© Glencoe/McGraw-Hill 734 Glencoe Geometry
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
© Glencoe/McGraw-Hill 735 Glencoe Geometry
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
© Glencoe/McGraw-Hill 736 Glencoe Geometry
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
Study Guide and Intervention (continued)
Volumes of Spheres
NAME ______________________________________________ DATE ____________ PERIOD _____
13-313-3
ExercisesExercises
ExampleExample
Skills PracticeVolumes of Spheres
NAME ______________________________________________ DATE ____________ PERIOD _____
13-313-3
© Glencoe/McGraw-Hill 737 Glencoe Geometry
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
© Glencoe/McGraw-Hill 738 Glencoe Geometry
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
© Glencoe/McGraw-Hill 739 Glencoe Geometry
Less
on
13-
3
Pre-Activity How can you find the volume of Earth?
Read the introduction to Lesson 13-3 at the top of page 702 in your textbook.
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.
Helping You Remember
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
© Glencoe/McGraw-Hill 740 Glencoe Geometry
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
© Glencoe/McGraw-Hill 741 Glencoe Geometry
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
© Glencoe/McGraw-Hill 742 Glencoe Geometry
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
Study Guide and Intervention (continued)
Congruent and Similar Solids
NAME ______________________________________________ DATE ____________ PERIOD _____
13-413-4
ExercisesExercises
ExampleExample
Skills PracticeCongruent and Similar Solids
NAME ______________________________________________ DATE ____________ PERIOD _____
13-413-4
© Glencoe/McGraw-Hill 743 Glencoe Geometry
Less
on
13-
4
Determine whether each pair of solids are similar, congruent, or neither.
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
© Glencoe/McGraw-Hill 744 Glencoe Geometry
Determine whether each pair of solids are similar, congruent, or neither.
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
© Glencoe/McGraw-Hill 745 Glencoe Geometry
Less
on
13-
4
Pre-Activity How are similar solids applied to miniature collectibles?
Read the introduction to Lesson 13-4 at the top of page 707 in your textbook.
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?
© Glencoe/McGraw-Hill 746 Glencoe Geometry
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
© Glencoe/McGraw-Hill 747 Glencoe Geometry
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
© Glencoe/McGraw-Hill 748 Glencoe Geometry
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)
Study Guide and Intervention (continued)
Coordinates in Space
NAME ______________________________________________ DATE ____________ PERIOD _____
13-513-5
ExercisesExercises
ExampleExample
Skills PracticeCoordinates in Space
NAME ______________________________________________ DATE ____________ PERIOD _____
13-513-5
© Glencoe/McGraw-Hill 749 Glencoe Geometry
Less
on
13-
5
Graph the rectangular solid that contains the given point and the origin asvertices. Label the coordinates of each vertex.
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.
Determine the distance between each pair of points. Then determine thecoordinates of the midpoint M of the segment joining the pair of points.
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)
© Glencoe/McGraw-Hill 750 Glencoe Geometry
Graph the rectangular solid that contains the given point and the origin asvertices. Label the coordinates of each vertex.
1. E(4, 6, %2) 2. R(%3, %5, 4)
Determine the distance between each pair of points. Then determine thecoordinates of the midpoint M of the segment joining the pair of points.
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
© Glencoe/McGraw-Hill 751 Glencoe Geometry
Less
on
13-
5
Pre-Activity How is three-dimensional graphing used in computer animation?
Read the introduction to Lesson 13-5 at the top of page 714 in your textbook.
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#
Helping You Remember
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
© Glencoe/McGraw-Hill 752 Glencoe Geometry
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
____
____
____
____
____
____
____
____
____
____
____
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ATE
____
____
____
PE
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D__
___
13-1
13-1
©G
lenc
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w-H
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3G
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Lesson 13-1
Vo
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spac
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at a
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.Vol
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cub
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that
mea
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s on
e un
it o
n ea
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cubi
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Volu
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If a
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)(3)
(4)
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3), h
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he p
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is 8
4 cu
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cent
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7 cm
3 cm4
cm
Fin
d t
he
volu
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of t
he
pri
sm i
f th
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f ea
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For
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!(6
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3.5)
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The
vol
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is 2
2.05
cub
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3.5
ft
base
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
volu
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ach
pri
sm.R
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d t
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512
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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
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cGra
w-H
ill72
4G
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eom
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Vo
lum
es o
f C
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der
sT
he v
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a c
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he p
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f th
e he
ight
and
the
are
a of
the
bas
e.T
he b
ase
of a
cyl
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r is
a c
ircl
e,so
the
are
a of
the
bas
e is
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.
Volu
me
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If a
cylin
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has
a vo
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Vcu
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units
, a h
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t of h
units
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Cyl
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run
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hen
V!
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r
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Stu
dy
Gu
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nte
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(con
tinu
ed)
Volu
mes
of P
rism
s an
d C
ylin
ders
NA
ME
____
____
____
____
____
____
____
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____
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____
____
____
PE
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13-1
13-1
Fin
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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)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
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
Answers (Lesson 13-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
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
Answers (Lesson 13-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
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
Answers (Lesson 13-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
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
Answers (Lesson 13-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
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
Answers (Lesson 13-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
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
Answers (Lesson 13-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
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
Answers (Lesson 13-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
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
Answers (Lesson 13-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
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
Answers (Lesson 13-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
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
Answers (Lesson 13-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
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
Answers (Lesson 13-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
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
Answers (Lesson 13-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
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
, 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
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ATE
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PE
RIO
D__
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13-5
13-5
Answers (Lesson 13-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Rea
din
g t
o L
earn
Math
emati
csC
oord
inat
es in
Spa
ce
NA
ME
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__D
ATE
____
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____
PE
RIO
D__
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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
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__D
ATE
____
____
____
PE
RIO
D__
___
13-5
13-5
Answers (Lesson 13-5)