Chapter 12 Resource Masters - Ms....

Chapter 12 Resource Masters

00i_GEOCRMC12_890521.indd 100i_GEOCRMC12_890521.indd 1 6/19/08 4:26:47 PM6/19/08 4:26:47 PM

Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.

ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4Homework Practice Workbook 0-07-890849-3 978-0-07-890849-1

Spanish VersionHomework Practice Workbook 0-07-890853-1 978-0-07-890853-8

Answers for Workbooks The answers for Chapter 12 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition text along with the English workbooks listed above.

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

Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9) These masters contain a Spanish version of Chapter 12 Test Form 2A and Form 2C.

Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 978-0-07-890521-6MHID: 0-07-890521-4

Printed in the United States of America.

6 7 8 9 10 11 12 REL 19 18 17 16 15 14 13

PDF Pass0ii_0vi_GEOCRMC12_890521.indd ii0ii_0vi_GEOCRMC12_890521.indd ii 3/5/13 10:25:54 PM3/5/13 10:25:54 PM

iii

ContentsTeachers Guide to Using the Chapter 12Resource Masters .............................................iv

Chapter ResourcesChapter 12 Student-Built Glossary .................... 1Chapter 12 Anticipation Guide (English) ........... 3Chapter 12 Anticipation Guide (Spanish) .......... 4

Lesson 12-1Representations of Three-Dimensional FiguresStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ..................................... 9Enrichment ...................................................... 10Graphing Calculator Activity ............................ 11

Lesson 12-2Surface Area of Prisms and CylindersStudy Guide and Intervention .......................... 12Skills Practice .................................................. 14Practice ............................................................ 15Word Problem Practice ................................... 16Enrichment ...................................................... 17

Lesson 12-3Surface Area of Pyramids and ConesStudy Guide and Intervention .......................... 18Skills Practice .................................................. 20Practice ............................................................ 21Word Problem Practice ................................... 22Enrichment ...................................................... 23Spreadsheet Activity ........................................ 24

Lesson 12-4Volumes of Prisms and CylindersStudy Guide and Intervention .......................... 25Skills Practice .................................................. 27Practice ............................................................ 28Word Problem Practice ................................... 29Enrichment ...................................................... 30

Lesson 12-5Volumes of Pyramids and ConesStudy Guide and Intervention .......................... 31Skills Practice .................................................. 33Practice ............................................................ 34Word Problem Practice ................................... 35Enrichment ...................................................... 36

Lesson 12-6Surface Areas and Volumes of SpheresStudy Guide and Intervention .......................... 37Skills Practice .................................................. 39Practice ............................................................ 40Word Problem Practice ................................... 41Enrichment ...................................................... 42

Lesson 12-7Spherical GeometryStudy Guide and Intervention .......................... 43Skills Practice .................................................. 45Practice ............................................................ 46Word Problem Practice ................................... 47Enrichment ...................................................... 48

Lesson 12-8Congruent and Similar SolidsStudy Guide and Intervention .......................... 49Skills Practice .................................................. 51Practice ............................................................ 52Word Problem Practice ................................... 53Enrichment ...................................................... 54

AssessmentStudent Recording Sheet ................................ 55Rubric for Extended-Response ....................... 56Chapter 12 Quizzes 1 and 2 ........................... 57Chapter 12 Quizzes 3 and 4 ........................... 58Chapter 12 Mid-Chapter Test .......................... 59Chapter 12 Vocabulary Test ........................... 60Chapter 12 Test, Form 1 ................................. 61Chapter 12 Test, Form 2A ............................... 63Chapter 12 Test, Form 2B ............................... 65Chapter 12 Test, Form 2C .............................. 67Chapter 12 Test, Form 2D .............................. 69Chapter 12 Test, Form 3 ................................. 71Chapter 12 Extended-Response Test ............. 73Standardized Test Practice ............................. 74

Answers ........................................... A1A36Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass00i_0vi_GEOCRMC12_890521.indd III00i_0vi_GEOCRMC12_890521.indd III 6/17/08 8:03:50 PM6/17/08 8:03:50 PM

iv

Teachers Guide to Using the Chapter 12 Resource Masters

The Chapter 12 Resource Masters includes the core materials needed for Chapter 12. These materials include worksheets, extensions, and assessment 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 on the TeacherWorks PlusTM CD-ROM.

Chapter ResourcesStudent-Built Glossary (pages 12) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 121. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 34) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.

Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. They are written for use with all levels of students.

Graphing Calculator, TI-Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.

Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass00i_0vi_GEOCRMC12_890521.indd IV00i_0vi_GEOCRMC12_890521.indd IV 6/17/08 8:04:05 PM6/17/08 8:04:05 PM

v

Assessment OptionsThe assessment masters in the Chapter 12 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended-Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 12 questions to assess students knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests Form 1 contains multiple-choice

questions and is intended for use with approaching grade level students.

Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

Form 3 is a free-response test for use with beyond grade level students.

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

Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

Full-size answer keys are provided for the assessment masters.

Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass0ii_0vi_GEOCRMC12_890521.indd v0ii_0vi_GEOCRMC12_890521.indd v 3/5/13 10:25:05 PM3/5/13 10:25:05 PM

PDF Pass00i_0vi_GEOCRMC12_890521.indd VI00i_0vi_GEOCRMC12_890521.indd VI 6/14/08 2:47:05 PM6/14/08 2:47:05 PM

Ch

apte

r R

eso

urc

es

Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD NAME DATE PERIOD

PDF Pass

Chapter 12 1 Glencoe Geometry

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

Student-Built Glossary12

Vocabulary TermFound

on PageDefi nition/Description/Example

altitude

axis

congruent solids

cross section

Euclidean geometry

great circle

isometric view

lateral area

lateral edge

lateral face

(continued on the next page)

001_025_GEOCRMC12_890521.indd 1001_025_GEOCRMC12_890521.indd 1 6/14/08 2:45:23 PM6/14/08 2:45:23 PM

Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


Vocabulary Term Found on Page Defi nition/Description/Example

oblique cone

oblique cylinder

oblique prism

regular pyramid

right cone

right cylinder

right prism (PRIZuhm)

similar solids

slant height

spherical geometry

Student-Built Glossary (continued)12


Ch

apte

r R

eso

urc

es

Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


PDF Pass


Before you begin Chapter 12

Read each statement.

Decide whether you Agree (A) or Disagree (D) with the statement.

Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 12

Reread each statement and complete the last column by entering an A or a D.

Did any of your opinions about the statements change from the first column?

For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

12 Anticipating GuideExtending Surface Area and Volume

Step 1

STEP 1A, D, or NS Statement

STEP 2A or D

1. The shape of a horizontal cross section of a square pyramid is a triangle.

2. The lateral area of a prism is equal to the sum of the areas of each face.

3. The axis of an oblique cylinder is different than the height of the cylinder.

4. The slant height and height of a regular pyramid are the same.

5. The lateral area of a cone equals the product of , the radius, and the height of the cone.

6. The volume of a right cylinder with radius r and height h is r2h.

7. The volume of a pyramid or a cone is found by multiplying the area of the base by the height.

8. To find the surface area of a sphere with radius r, multiply r2 by 4.

9. All postulates and properties of Euclidean geometry are true in spherical geometry.

10. All spheres and all cubes are similar solids.

Step 2


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NOMBRE FECHA PERODO

PDF 2nd

Antes de comenzar el Captulo 12

Lee cada enunciado.

Decide si ests de acuerdo (A) o en desacuerdo (D) con el enunciado.

Escribe A o D en la primera columna O si no ests seguro(a) de la respuesta, escribe NS (No estoy seguro(a).

Despus de completar el Captulo 12

Vuelve a leer cada enunciado y completa la ltima columna con una A o una D.

Cambi cualquiera de tus opiniones sobre los enunciados de la primera columna?

En una hoja de papel aparte, escribe un ejemplo de por qu ests en desacuerdo con los enunciados que marcaste con una D.

12 Ejercicios PreparationsExtiende el rea de Superficie y volumen

Paso 1

PASO 1A, D o NS Enunciado

PASO 2A o D

1. La forma de un corte transversal horizontal de una pirmide cuadrada es un tringulo.

2. El rea lateral de un prisma es igual a la suma de las reas de cada cara.

3. El eje de un cilindro oblicuo es diferente a la altura del cilindro.

4. La altura oblicua y la altura de una pirmide regular son las mismas.

5. El rea lateral de un cono es igual al producto de , el radio, por la altura del cono.

6. El volumen de un cilindro recto con radio r y altura h es r2h.

7. El volumen de una pirmide o un cono se calcula multiplicando el rea de la base por la altura.

8. Para calcular el rea de superficie de una esfera con radio r, multiplica r2 por 4.

9. Todos los postulados y propiedades de la geometra euclidiana son verdaderos en geometra esfrica.

10. Todas las esferas y todos los cubos son slidos semejantes.

Paso 2

Captulo 12 4 Geometra de Glencoe


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12

-1

PDF Pass


Draw Isometric Views Isometric dot paper can be used to draw isometric views, or corner views, of a three-dimensional object on two-dimensional paper.

Use isometric dot paper to sketch a triangular prism 3 units high, with two sides of the base that are 3 unitslong and 4 units long.Step 1 Draw

AB at 3 units and draw

AC at 4 units.Step 2 Draw

AD ,

BE , and

CF , each at 3 units.Step 3 Draw

BC and DEF.

Use isometric dot paper and the orthographic drawing to sketch a solid. The top view indicates two columns. The right and left views indicate that the height of figure is

three blocks. The front view indicates that the columns have heights 2 and 3 blocks.

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of each column.

ExercisesSketch each solid using isometric dot paper.

1. cube with 4 units on each side 2. rectangular prism 1 unit high, 5 units long, and 4 units wide

Use isometric dot paper and each orthographic drawing to sketch a solid.

3.

top view left view front view right view

4.


Study Guide and InterventionRepresentations of Three-Dimensional Figures

12-1

Example 1

A

BC

D

E F

Example 2


object


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


Cross Sections The intersection of a solid and a plane is called a cross section of the solid. The shape of a cross section depends upon the angle of the plane.

There are several interesting shapes that are cross sections of a cone. Determine the shape resulting from each cross section of the cone.

a. If the plane is parallel to the base of the cone, then the resulting cross section will be a circle.

b. If the plane cuts through the cone perpendicular to the base and through the center of the cone, then the resulting cross section will be a triangle.

c. If the plane cuts across the entire cone, thenthe resulting cross section will be an ellipse.

ExercisesDescribe each cross section.

1. 2. 3.

Study Guide and Intervention (continued)Representations of Three-Dimensional Figures

12-1

Example


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

1

PDF Pass


12-1

Use isometric dot paper to sketch each prism.

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


3. 4.

Describe each cross section.

5. 6.

7. 8.

Skills PracticeRepresentations of Three-Dimensional Figures

top view left view front view right viewtop view left view front view right view

001_025_GEOCRMC12_890521.indd 7001_025_GEOCRMC12_890521.indd 7 6/1/09 11:09:44 AM6/1/09 11:09:44 AM

Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


Use isometric dot paper to sketch each prism.

1. rectangular prism 3 units high, 2. triangular prism 3 units high, whose bases 3 units long, and 2 units wide are right triangles with legs 2 units and 4 units long


3.


4.


Sketch the cross section from a vertical slice of each figure.

5. 6.

7. SPHERES Consider the sphere in Exercise 5. Based on the cross section resulting from a horizontal and a vertical slice of the sphere, make a conjecture about all spherical cross sections.

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

PracticeRepresentations of Three-Dimensional Figures

12-1


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

1

PDF Pass


1. LABELS Jamal removes the label from a cylindrical soup can to earn points for his school. Sketch the shape of the label.

2. BLOCKS Margots three-year-old son made the magnetic block sculpture shown below in corner view.

Draw the right view of the sculpture.

3. CUBES Nathan marks the midpoints of three edges of a cube as shown.He then slices the cube along a plane that contains these three points. Describe the resulting cross section.

4. ENGINEERING Stephanie needs an object whose top view is a circle and whose left and front views are squares. Describe an object that will satisfy these conditions.

5. DESK SUPPORTS The figure shows the support for a desk.

a. Draw the top view.

b. Draw the front view.

c. Draw the right view.

Word Problem PracticeRepresentations of Three-Dimensional Figures

12-1


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


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

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

1. 2.

3. 4.

5. 6.

Enrichment12-1


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

1

PDF Pass


The science of perspective drawing studies how to draw a three-dimensional object on a two-dimensional page. This science became highly refined during the Renaissance with the work of artists such as Albrecht Drer and Leonardo da Vinci.

Today, computers are often used to make perspective drawings, particularly elaborate graphics used in television and movies. The three-dimensional coordinates of objects are figured. Then algebra is used to transform these into two-dimensional coordinates. The graph of these new coordinates is called a projection.

The formulas below will draw one type of projection in which the y-axis is drawn horizontally, the z-axis vertically, and the x-axis at an angle of a with the y-axis. If the three-dimensional coordinates of a point are (x, y, z), then the projection coordinates (X, Y) are given by

X = x(-cos a) + y and Y = x(-sin a) + z.Although this type of projection gives a fairly good perspective drawing, it does distort some lengths.

1. The drawing with the coordinates given below is a cube. A(5, 0, 5), B(5, 5, 5), C(5, 5, 0), D(5, 0, 0),

E(0, 0, 5), F(0, 5, 5), G(0, 5, 0), H(0, 0, 0) Use the formulas above to find the projection coordinates of each

point, using a = 45. Round projection coordinates to the nearest integer. Graph the cube on a graphing calculator. Make a sketch of the display.

A'(__, __) B'(__, __) C(__, __) D(__, __) E'(__, __) F'(__, __) G(__, __) H(__, __)

2. The points A(10, 2, 0), B(10, 10, 0), C(2, 10, 0), and D(3, 3, 4) are vertices of a pyramid. Find the projection coordinates, using a = 25. Round coordinates to the nearest integer. Then graph the pyramid on a graphing calculator by drawing

AB ,

BC ,

CD ,

DA , and

DB . Make a sketch of the display. A(__, __) B(__, __) C(__, __) D(__, __)

Graphing Calculator ActivityPerspective Drawings

12-1

E F

A B

D

H

C

G


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


Study Guide and InterventionSurface Areas of Prisms and Cylinders

12-2

Lateral and Surface Areas of Prisms In a solid figure, faces that are not bases are lateral faces. The lateral area is the sum of the area of the lateral faces. The surface area is the sum of the lateral area and the area of the bases.

Lateral Area of a Prism

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

Surface Area of a Prism

If a prism has a surface area of S square units, a lateral area of L square units, and each base has an area of B square units, then S = L + 2B or S = Ph + 2B

Find the lateral and surface area of the regular pentagonal prism above if each base has a perimeter of 75 centimeters and the height is 10 centimeters.

L = Ph Lateral area of a prism = 75(10) P = 75, h = 10 = 750 Multiply.

The lateral area is 750 square centimeters and the surface area is about 1524.2 square centimeters.

ExercisesFind the lateral area and surface area of each prism. Round to the nearest tenth if necessary.

1.

4 m

3 m10 m

2.

15 in.

10 in.

8 in.

3.

6 in.18 in.

4.

20 cm

10 cm 10 cm

12 cm8 cm9 cm

5.

4 in.

4 in.

12 in.

6.

4 m16 m

Example

pentagonal prism

altitude

lateraledge lateral

face

S = L + 2B = 750 + 2 ( 1 2 aP) = 750 + ( 7.5 tan 36 ) (75) 1524.2

tan 36 = 7.5 a

a = 7.5 tan 36

a

15 cm

36


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

2

PDF Pass


Lateral and Surface Areas of Cylinders A cylinder is a solid with bases that are congruent circles lying in parallel planes. The axis of a cylinder is the segment with endpoints at the centers of these circles. For a right cylinder, the axis is also the altitude of the cylinder.

Lateral Area of a Cylinder

If a cylinder has a lateral area of L square units, a height of h units, and a base has a radius of r units, then L = 2rh.

Surface Area of a Cylinder

If a cylinder has a surface area of S square units, a height of h units, and a base has a radius of r units, then S = L + 2B or 2rh + 2r2.

Find the lateral and surface area of the cylinder. Round to the nearest tenth.If d = 12 cm, then r = 6 cm.L = 2rh Lateral area of a cylinder = 2(6)(14) r = 6, h = 14 527.8 Use a calculator.

S = 2rh + 2r2 Surface area of a cylinder 527.8 + 2(6)2 2rh 527.8, r = 6 754.0 Use a calculator.The lateral area is about 527.8 square centimeters and the surface area is about 754.0 square centimeters.

ExercisesFind the lateral area and surface area of each cylinder. Round to the nearest tenth. 1.

12 cm

4 cm 2.

6 in.10 in.

3.

6 cm

3 cm

3 cm

4.

20 cm

8 cm

5.

12 m

4 m

6. 2 m1 m

radius of baseaxis

base

baseheight

Example

14 cm

12 cm

12-2 Study Guide and Intervention (continued)Surface Areas of Prisms and Cylinders


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


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

1. 2.

3. 4.

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

5. 6.

7. 8.

12 yd

12 yd

10 yd8 m

6 m

12 m

10 in.5 in.

6 in.

8 in.

9 cm

9 cm

7.8 cm9 cm

12 cm

Skills PracticeSurface Areas of Prisms and Cylinders

12-2

8 in.

12 in.

2 yd

3 yd

12 in.

10 in.

2 m

2 m


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

2

PDF Pass


12-2 PracticeSurface Areas of Prisms and Cylinders

Find the lateral and surface area of each prism. Round to the nearest tenth if necessary. 1.

15 cm

15 cm

32 cm

2.

8 ft

10 ft5 ft

3.

2 m11 m

4.

4 yd

4 yd

9.5 yd

5 yd

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

5. 5 ft

7 ft

6. 4 m

8.5 m

7. 19 in.

17 in.

8.

12 m30 m


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


1. LOGOS The Z company specializes in caring for zebras. They want to make a 3-dimensional Z to put in front of their company headquarters. The Z is 15 inches thick and the perimeter of the base is 390 inches.

15"

What is the lateral surface area of this Z?

2. STAIRWELLS Management decides to enclose stairs connecting the first and second floors of a parking garage in a stairwell shaped like an oblique rectangular prism.

16 ft

20 ft

15 ft

9 ft

What is the lateral surface area of the stairwell?

3. CAKES A cake is a rectangular prism with height 4 inches and base 12 inches by 15 inches. Wallace wants to apply frosting to the sides and the top of the cake. What is the surface area of the part of the cake that will have frosting?

4. EXHAUST PIPES An exhaust pipe is shaped like a cylinder with a height of 50 inches and a radius of 2 inches. What is the lateral surface area of the exhaust pipe? Round your answer to the nearest hundredth.

5. TOWERS A circular tower is made by placing one cylinder on top of another. Both cylinders have a height of 18 inches. The top cylinder has a radius of 18 inches and the bottom cylinder has a radius of 36 inches.

18 in.

18 in.

a. What is the total surface area of the tower? Round your answer to the nearest hundredth.

b. Another tower is constructed by placing the original tower on top of another cylinder with a height of 18 inches and a radius of 54 inches. What is the total surface area of the new tower? Round your answer to the nearest hundredth.

12-2 Word Problem PracticeSurface Areas of Prisms and Cylinders


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

2

PDF Pass


Enrichment12-2

Minimizing Cost in ManufacturingSuppose that a manufacturer wants to make a can that has a volume of 40 cubic inches. The cost to make the can is 3 cents per square inch for the top and bottom and 1 cent per square inch for the side.

1. Write the value of h in terms of r, given v = r2h.

2. Write a formula for the cost in terms of r.

3. Use a graphing calculator to graph the formula, letting Y1 represent the cost and X represent r. Use the graph to estimate the point at which the cost is minimized.

4. Repeat the procedure using 2 cents per square inch for the top and bottom and 4 cents per square inch for the top and bottom.

5. What would you expect to happen as the cost of the top and bottom increases?

6. Compute the table for the cost value given. What happens to the height of the can as the cost of the top and bottom increases?

Cost Top Cost Minimum

& Bottom Cylinder h

2 cents 1 cent

3 cents 1 cent

4 cents 1 cent

5 cents 1 cent

6 cents 1 cent

r

h


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


12-3 Study Guide and InterventionSurface Areas of Pyramids and Cones

Lateral and Surface Areas of Pyramids A pyramid is a solid with a polygon base. The lateral faces intersect in a common point known as the vertex. The altitude is the segment from the vertex that is perpendicular to the base. For a regular pyramid, the base is a regular polygon and the altitude has an endpoint at the center of the base. All the lateral edges are congruent and all the lateral faces are congruent isosceles triangles. The height of each lateral face is called the slant height.

Lateral Area of a Regular Pyramid

The lateral area L of a regular pyramid is L = 1 2 P, where

is the slant height and P is the perimeter of the base.

Surface Area of a Regular Pyramid

The surface area S of a regular pyramid is S = 1 2 P + B,

where is the slant height, P is the perimeter of the base, and B is the area of the base.

For the regular square pyramid above, find the lateral area and surface area if the length of a side of the base is 12 centimeters and the height is 8 centimeters. Round to the nearest tenth if necessary.

Find the slant height.2 = 62 + 82 Pythagorean Theorem2 = 100 Simplify. = 10 Take the positive square root of each side.

The lateral area is 240 square centimeters, and the surface area is 384 square centimeters.

ExercisesFind the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

1.

15 cm

20 cm

2.

45

8 ft

3.

60

10 cm 4.

6 in.8.7 in. 15 in.

lateral edge

baseslant height height

Example

L = 1 2 P Lateral area of a regular pyramid

= 1 2 (48)(10) P = 4 12 or 48, = 10

= 240 Simplify.

S = 1 2 P + B Surface area of a regular pyramid

= 240 + 144 1 2 P = 240, B = 12 12 or 144

= 384


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

3

PDF Pass


12-3 Study Guide and Intervention (continued)Surface Areas of Pyramids and Cones

Lateral and Surface Areas of Cones A cone has a circular base and a vertex. The axis of the cone is the segment with endpoints at the vertex and the center of the base. If the axis is also the altitude, then the cone is a right cone. If the axis is not the altitude, then the cone is an oblique cone.

Lateral Area of a Cone

The lateral area L of a right circular cone is L = r, where r is the radius and is the slant height.

Surface Area of a Cone

The surface area S of a right cone is S = r + r2, where r is the radius and is the slant height.

For the right cone above, find the lateral area and surface area if the radius is 6 centimeters and the height is 8 centimeters. Round to the nearest tenth if necessary.

Find the slant height.2 = 62 + 82 Pythagorean Theorem2 = 100 Simplify. = 10 Take the positive square root of each side.

The lateral area is about 188.5 square centimeters and the surface area is about 301.6 square centimeters.

ExercisesFind the lateral area and surface area of each cone. Round to the nearest tenth if necessary.

1.

9 cm

12 cm 2.

5 ft

30

3. 12 cm13 cm

4.

4 in.

45

axis

base base

slant height

right coneoblique cone

altitudeV V

Example

L = r Lateral area of a right cone = (6)(10) r = 6, = 10 188.5 Simplify.

S = r + r2 Surface area of a right cone 188.5 + (62) r 188.5, r = 6 301.6 Simplify.


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


12-3 Skills PracticeSurface Areas of Pyramids and Cones

Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

1. 2.

3. 4.

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

5. 6.

7. 8.

4 cm

7 cm20 in.

8 in.

9 m

10 m 14 ft

12 ft

14 m

5 m 10 ft

25 ft

8 in.

21 in.

17 mm

9 mm


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

3

PDF Pass


12-3 Practice Surface Areas of Pyramids and Cones

Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary.

1.

9 yd

10 yd

2. 12 m

7 m

3.

13 ft

5 ft

4.

8 cm

2.5 cm

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

5.

5 m4 m

6. 7 cm

21 cm

7. Find the surface area of a cone if the height is 14 centimeters and the slant height is 16.4 centimeters.

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

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

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


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


12-3 Word Problem PracticeSurface Areas of Pyramids and Cones

1. PAPER MODELS Patrick is making a paper model of a castle.Part of the model involves cutting out the net shown and folding it into a pyramid. The pyramid has a square base. What is the lateral surface area of the resulting pyramid?

2. TETRAHEDRON Sung Li builds a paper model of a regular tetrahedron, a pyramid with an equilateral triangle for the base and three equilateral triangles for the lateral faces. One of the faces of the tetrahedron has an area of 17 square inches. What is the total surface area of the tetrahedron?

3. PAPERWEIGHTS Daphne uses a paperweight shaped like a pyramid with a regular hexagon for a base. The side length of the regular hexagon is 1 inch. The altitude of the pyramid is 2 inches.

What is the lateral surface area of this pyramid? Round your answers to the nearest hundredth.

4. SPRAY PAINT A can of spray paint shoots out paint in a cone shaped mist. The lateral surface area of the cone is 65 square inches when the can is held 12 inches from a canvas. What is the area of the part of the canvas that gets sprayed with paint? Round your answer to the nearest hundredth.

5. MEGAPHONES A megaphone is formed by taking a cone with a radius of 20 centimeters and an altitude of 60 centimeters and cutting off the tip. The cut is made along a plane that is perpendicular to the axis of the cone and intersects the axis 12 centimeters from the vertex. Round your answers to the nearest hundredth.

a. What is the lateral surface area of the original cone?

b. What is the lateral surface area of the tip that is removed?

c. What is the lateral surface area of the megaphone?

20 cm20 cm15 cm


Less

on

12-

3

Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


PDF Pass


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

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

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

3. What is the circumference of the complete circle?

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

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

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

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

8. What is the slant height of the cone?

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

10. Find the lateral area.

11. Find the total surface area.

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

12.

1206 cm

13.

20 cm

diameter of base = diameter of base =

lateral area = lateral area =

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

Enrichment12-3


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.


PDF Pass


12-3

You can use a spreadsheet to determine the surface area of a cone.

Lucy wants to wrap a Mothers Day gift. The gift she has bought for her mother is in a conical box that has a slant height of 6 inches and has a radius of 3 inches. She must determine the surface area of the box to determine how much wrapping paper to buy. Use a spreadsheet to determine the surface area of the box. Round to the nearest tenth.

Step 1 Use cell A1 for the radius of the cone and cell B1 for the height.

Step 2 In cell C1, enter an equals sign followed by PI()*A1*B1 + PI()*A1^2. Then press ENTER. This will return the surface area of the cone.

The surface area of the conical box is 84.8 in2 to the nearest tenth.

Use a spreadsheet to determine the surface area of a cone that has a radius of 2.5 centimeters and a slant height of 5.2 centimeters. Round to the nearest tenth.

Step 1 Use cell A2 for the radius of the cone and cell B2 for the slant height.

Step 2 Click on the bottom right corner of cell C1 and drag it to C2. This returns the surface area of the cone.

The surface area of the cone is 60.5 cm2 to the nearest tenth.

ExercisesUse a spreadsheet to find the surface area of each cone with the given dimensions. Round to the nearest tenth.

1. r = 12 m, = 2.3 m 2. r = 6 m, = 2 m

3. r = 3 in., = 7 in. 4. r = 5 in., = 11 in.

5. r = 1 ft, = 3 ft 6. r = 3 ft, = 1.5 ft

7. r = 10 mm, = 20 mm 8. r = 1.5 mm, = 4.5 mm

9. r = 6.2 cm, = 1.2 cm 10. r = 10 cm, = 15 cm

11. r = 10 m, = 2 m 12. r = 11 m, = 13 m

Spreadsheet ActivitySurface Areas of Cones

Example 1

Example 2

A12

B C

Sheet 1


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.


Less

on

12-

4

PDF Pass


Study Guide and InterventionVolumes of Prisms and Cylinders

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.

Volume of a Prism

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

Find the volume of the prism.

7 cm3 cm

4 cm

V = Bh Volume of a prism = (7)(3)(4) B = (7)(3), h = 4 = 84 Multiply.The volume of the prism is 84 cubic centimeters.

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

3.5 ft

base

V = Bh Volume of a prism = (6.3)(3.5) B = 6.3, h = 3.5 = 22.05 Multiply.The volume is 22.05 cubic feet.

Exercises

Find the volume of each prism.

1.

8 ft

8 ft

8 ft

2.

3 cm

4 cm

1.5 cm

3.

3015 ft

12 ft 4.

10 ft15 ft

12 ft

5.

4 cm

6 cm

2 cm

1.5 cm

6.

7 yd4 yd

3 yd

12-4

Example 1 Example 2

cubic foot cubic yard27 cubic feet = 1 cubic yard


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Study Guide and Intervention (continued)Volumes of Prisms and Cylinders

Volumes of Cylinders The volume of a cylinder is the product of the height and the area of the base. When a solid is not a right solid, use Cavalieris Priniciple to find the volume. The principle states that if two solids have the same height and the same cross sectional area at every level, then they have the same volume.

Find the volume of the cylinder.

4 cm

3 cm

V = r2h Volume of a cylinder = (3)2(4) r = 3, h = 4 113.1 Simplify.The volume is about 113.1 cubic centimeters.

Find the volume of the oblique cylinder.

8 in.

13 in.

5 in.

h

Use the Pythagorean Theorem to find the height of the cylinder.h2 + 52 = 132 Pythagorean Theorem h2 = 144 Simplify. h = 12 Take the positive square root of each side.

V = r2h Volume of a cylinder = (4)2(12) r = 4, h = 12 603.2 Simplify.The Volume is about 603.2 cubic inches.

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

1. 2 ft

1 ft

2. 18 cm

2 cm

3.

12 ft1.5 ft

4. 20 ft

20 ft

5.

10 cm

13 cm

6.

1 yd4 yd

12-4

Volume of a Cylinder

If a cylinder has a volume of V cubic units, a height of h units, and the bases have a radius of r units, then V = r 2h.

Example 1 Example 2

r

h


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

4

PDF Pass


12-4

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

1.

18 cm

16 cm

8 cm 2.

6 ft

8 ft

2 ft

3.

3 m

5 m

13 m

4.

16 in. 22 in.

34 in.

5.

15 mm23 mm

6. 6 yd

10 yd

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

7. 8.

5 in.

3 in.

Skills PracticeVolumes of Prisms and Cylinders

17 cm

18 cm

4 cm


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


PracticeVolumes of Prisms and Cylinders

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

1.

17 m10 m

26 m 2.

5 in.

5 in.

5 in.

9 in.

3.

16 mm 17.5 mm

4. 7 ft 25 ft

5.

13 yd

20 yd

10 yd 6.

30 cm

8 cm

7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25 1

2 inches and a radius of 21 inches.

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

b. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium hold?

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

12-4


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

4

PDF Pass


Word Problem PracticeVolumes of Prisms and Cylinders

1. TRASH CANS The Meyer family uses a kitchen trash can shaped like a cylinder. It has a height of 18 inches and a base diameter of 12 inches.What is the volume of the trash can? Round your answer to the nearest tenth of a cubic inch.

2. BENCH Inside a lobby, there is a piece of furniture for sitting. The furniture is shaped like a simple block with a square base 6 feet on each side and a height of 1 3

5 feet.

6 ft6 ft

1 ft35

What is the volume of the seat?

3. FRAMES Margaret makes a square frame out of four pieces of wood. Each piece of wood is a rectangular prism with a length of 40 centimeters, a height of 4 centimeters, and a depth of 6 centimeters. What is the total volume of the wood used in the frame?

4. PENCIL GRIPS A pencil grip is shaped like a triangular prism with a cylinder removed from the middle. The base of the prism is a right isosceles triangle with leg lengths of 2 centimeters. The diameter of the base of the removed cylinder is 1 centimeter. The heights of the prism and the cylinder are the same, and equal to 4 centimeters.

What is the exact volume of the pencil grip?

5. TUNNELS Construction workers are digging a tunnel through a mountain. The space inside the tunnel is going to be shaped like a rectangular prism. The mouth of the tunnel will be a rectangle 20 feet high and 50 feet wide and the length of the tunnel will be 900 feet.

a. What will the volume of the tunnel be?

b. If instead of a rectangular shape, the tunnel had a semicircular shape with a 50-foot diameter, what would be its volume? Round your answer to the nearest cubic foot.

18 in.

12 in.

12-4


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Visible Surface AreaUse 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 surface covered by other cubes or by the table or desk.

1. 2.

volume = volume =

visible surface area = visible surface area =

3. 4. 5.

volume = volume = volume =

visible surface area = visible surface area = visible surface area =

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

volume =

visible surface area =

4 in.

4 in.

3 in.

8 in.

3 in.

5 in.

5 in.

3 in.

3 in.

12-4 Enrichment


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

5

PDF Pass


Study Guide and InterventionVolumes of Pyramids and Cones

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.

Find the volume of the square pyramid.

V = 1 3 Bh Volume of a pyramid

= 1 3 (8)(8)10 B = (8)(8), h = 10

213.3 Multiply.The volume is about 213.3 cubic feet.

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

1.

12 ft

8 ft

10 ft 2.

10 ft

6 ft15 ft

3.

4 cm8 cm

12 cm 4.

18 ft

regularhexagon 6 ft

5.

15 in.

15 in.

16 in. 6. 6 yd

8 yd

5 yd

8 ft

8 ft

10 ft

12-5

Volume of a Pyramid

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

3 Bh.

Example


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


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

Find the volume of the cone.

V = 1 3 r2h Volume of a cone

= 1 3 (5)212 r = 5, h = 12

314.2 Simplify.The volume of the cone is about 314.2 cubic centimeters.

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

1.

6 cm10 cm

2. 8 ft

10 ft

3.

30 in.

12 in.

4. 4518 yd

20 yd

5. 26 ft20 ft

6.

16 cm

45

12 cm

5 cm

Study Guide and Intervention (continued)Volumes of Pyramids and Cones

12-5

Volume of a Cone

If a cone has a volume of V cubic units, a height of h units, and the bases have a radius of r units, then V = 1

3 r2h.

Example

r

h


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

5

PDF Pass


12-5 Skills PracticeVolumes of Pyramids and Cones

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 if necessary.

7. 8.

5 ft5 ft

8 ft

4 cm7 cm

8 cm

8 in.10 in.

14 in.25 m

12 m

25 yd

14 yd66

18 mm

4 ft4 ft

6 ft12 cm

6 cm


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


PracticeVolumes of Pyramids and Cones

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

1.

9.2 yd9.2 yd

13 yd

2.

12.5 cm25 cm

23 cm

3.

19 ft

9 ft 4.

5212 mm

5.

6 in.6 in.

11 in.

6.

37 ft11 ft

7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 meters in 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 Zosers pyramid, erected in the 27th century B.C. In its original state, it stood 62 meters high with a rectangular base that measured 140 meters by 118 meters. Find the volume of the original pyramid.

12-5


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

5

PDF Pass


1. ICE CREAM DISHES The part of a dish designed for ice cream is shaped like an upside-down cone. The base of the cone has a radius of 2 inches and the height is 1.2 inches.

What is the volume of the cone? Round your answer to the nearest hundredth.

2. GREENHOUSES A greenhouse has the shape of a square pyramid. The base has a side length of 30 yards. The height of the greenhouse is 18 yards.

18

yd

30 yd

What is the volume of the greenhouse?

3. TEEPEE Caitlyn made a teepee for a class project. Her teepee had a diameter of 6 feet. The angle the side of the teepee made with the ground was 65.

65

What was the volume of the teepee? Round your answer to the nearest hundredth.

4. SCULPTING A sculptor wants to remove stone from a cylindrical block 3 feet high and turn it into a cone. The diameter of the base of the cone and cylinder is 2 feet.

What is the volume of the stone that the sculptor must remove? Round your answer to the nearest hundredth.

5. STAGES A stage has the form of a square pyramid with the top sliced off along a plane parallel to the base. The side length of the top square is 12 feet and the side length of the bottom square is 16 feet. The height of the stage is 3 feet.

12 feet

16 feet

3 feet

a. What is the volume of the entire square pyramid that the stage is part of?

b. What is the volume of the top of the pyramid that is removed to get the stage?

c. What is the volume of the stage?

Word Problem PracticeVolumes of Pyramids and Cones

12-5


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Enrichment

FrustumsA frustum is a figure formed when a plane intersects a pyramid or cone so that the plane is parallel to the solids base. The frustum is the part of the solid between the plane and the base. To find the volume of a frustum, the areas of both bases must be calculated and used in the formula.

V = 1 3 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. 13 cm6 cm

9 cm

5 cm

19.5 cm

2.

7.5 in.

4.5 in.

3 in.

3.

8 m

6 m

12 m

4.5 m2.25 m

3 m

5 m

4.

12 ft13 ft

7 ft

12-5


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

6

PDF Pass


Study Guide and InterventionSurface Areas and Volumes of Spheres

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

Find the surface area of a sphere to the nearest tenth if the radius of the sphere is 6 centimeters.

S = 4r2 Surface area of a sphere = 4(6)2 r = 6 452.4 Simplify.

The surface area is 452.4 square centimeters.

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

1. 5 m

2.

7 in

3.

3 ft

4.

9 cm

5. sphere: circumference of great circle = cm

6. hemisphere: area of great circle 4 ft2

r

6 cm

Surface Area of a Sphere If a sphere has a surface area of S square units and a radius of r units, then S = 4r

2.

Example

12-6


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Study Guide and Intervention (continued)Surface Areas and Volumes of Spheres

12-6

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

Find the volume of a sphere with radius 8 centimeters.

V = 4 3 r3 Volume of a sphere

= 4 3 (8)3 r = 8

2144.7 Simplify.The volume is about 2144.7 cubic centimeters.

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

1.

5 ft

2. 6 in. 3.

16 in.

4. hemisphere: radius 5 in.

5. sphere: circumference of great circle 25 ft

6. hemisphere: area of great circle 50 m2

r

8 cm

Volume of a Sphere

If a sphere has a volume of V cubic units and a radius of r units, then V = 4 3 r3.

Example


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

6

PDF Pass


Skills PracticeSurface Areas and Volumes of Spheres

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

1.

7 in.

2.

32 m

3. hemisphere: radius of great circle = 8 yd

4. sphere: area of great circle 28.6 in2

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

5.

16.2 cm

6.

94.8 ft

7. hemisphere: diameter = 48 yd

8. sphere: circumference of a great circle 26 m

9. sphere: diameter = 10 in.

12-6


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


12-6 PracticeSurface Areas and Volumes of Spheres

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

1.

6.5 cm

2.

89 ft

3. hemisphere: radius of great circle = 8.4 in.

4. sphere: area of great circle 29.8 m2

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

5.

12.32 ft

6.32 m

7. hemisphere: diameter = 18 mm

8. sphere: circumference 36 yd

9. sphere: radius = 12.4 in.


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

6

PDF Pass


Word Problem PracticeSurface Areas and Volumes of Spheres

1. ORANGES Mandy cuts a spherical orange in half along a great circle. If the radius of the orange is 2 inches, what is the area of the cross section that Mandy cut? Round your answer to the nearest hundredth.

2. BILLIARDS A billiard ball set consists of 16 spheres, each 2 1

4 inches in

diameter. What is the total volume of a complete set of billiard balls? Round your answer to the nearest thousandth of a cubic inch.

3. MOONS OF SATURN The planet Saturn has several moons. These can be modeled accurately by spheres. Saturns largest moon Titan has a radius of about 2575 kilometers. What is the approximate surface area of Titan? Round your answer to the nearest tenth.

4. THE ATMOSPHERE About 99% of Earths atmosphere is contained in a 31-kilometer thick layer that enwraps the planet. The Earth itself is almost a sphere with radius 6378 kilometers. What is the ratio of the volume of the atmosphere to the volume of Earth? Round your answer to the nearest thousandth.

5. CUBES Marcus builds a sphere inside ofa cube. The sphere fits snugly inside the cube so that the sphere touches the cube at one point on each side. The side length of the cube is 2 inches.

a. What is the surface area of the cube?

b. What is the surface area of the sphere? Round your answers to the nearest hundredth.

c. What is the ratio of the surface area of the cube to the surface area of the sphere? Round your answer to the nearest hundredth.

12-6


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Enrichment

Spheres and DensityThe density of a metal is a ratio of its mass to its volume. For example, 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 4 3 (0.4)3

10.5(0.27) 2.81

The mass is about 2.81 grams.

ExercisesFind 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 grams. Find the radius of the ball to the nearest tenth of a centimeter.

6. An iron ball weighs 804 grams. 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 centimeters. Which weighs more? How much more?

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

12-6

Example


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

7

PDF Pass


12-7 Study Guide and InterventionSpherical Geometry

Geometry On A Sphere Up to now, we have been studying Euclidean geometry, where a plane is a flat surface made up of points that extends infinitely in all directions. In spherical geometry, a plane is the surface of a sphere.

Name each of the following on sphere K.

a. two lines containing the point F

EG and BH are lines on sphere K that contain the point F

b. a line segment containing the point J

ID is a segment on sphere K that contains the point J

c. a triangle

AHI is a triangle on sphere K

ExercisesName two lines containing point Z, a segment containing point R, and a triangle in each of the following spheres.

1.

F

2.

M

Determine whether figure u on each of the spheres shown is a line in spherical geometry.

3. 4.

5. GEOGRAPHY Lines of latitude run horizontally across the surface of Earth. Are there any lines of latitude that are great circles? Explain.

Example

K


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Comparing Euclidean and Spherical Geometries Some postulates and properties of Euclidean geometry are true in spherical geometry. Others are not true or are true only under certain circumstances.

Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.

Given any line, there are an infinite number of parallel lines.

On the sphere to the right, if we are given line m we see that it goes through the poles of the sphere. If we try to make any other line on the sphere, it would intersect line m at exactly 2 points. This property is not true in spherical geometry.A corresponding statement in spherical geometry would be: Given any line, there are no parallel lines.

ExercisesTell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.

1. If two nonidentical lines intersect at a point, they do not intersect again.

2. Given a line and a point on the line, there is only one perpendicular line going through that point.

3. Given two parallel lines and a transversal, alternate interior angles are congruent.

4. If two lines are perpendicular to a third line, they are parallel.

5. Three noncollinear points determine a triangle.

6. A largest angle of a triangle is opposite the largest side.

12-7 Study Guide and Intervention (continued)Spherical Geometry

Example


Less

on

12-

7

Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

PDF Pass


Name two lines containing point K, a segment containing point T, and a triangle in each of the following spheres.

1.

C

2.

L


3. 4.

basketball

Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning.

5. If two lines form vertical angles, then the angles are equal in measure.

6. If two lines meet a third line at the same angle, those lines are parallel.

7. Two lines meet at two 90 angles or they meet at angles whose sum is 180.

8. Three non-parallel lines divide the plane into 7 separate parts.

Skills PracticeSpherical Geometry

12-7


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Name two lines containing point K, a segment containing point T, and a triangle in each of the following spheres.

1.

L

2.

M


3.

tennis ball

4.

Tell whether the following postulate or property of plane Euclidean geometry has a corresponding statement in spherical geometry. If so, write the corresponding statement. If not, explain your reasoning. 5. A triangle can have at most one obtuse angle.

6. The sum of the angles of a triangle is 180.

7. Given a line and a point not on the line, there is exactly one line that goes through the point and is perpendicular to the line.

8. All equilateral triangles are similar.

9. AIRPLANES When flying an airplane from New York to Seattle, what is the shortest route: flying directly west, or flying north across Canada? Explain.

12-7 PracticeSpherical Geometry


Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

Less

on

12-

7

PDF Pass


12-7 Word Problem PracticeSpherical Geometry

1. PAINTING Consider painting quadrilateral ABCD on the beach ball with radius 1 ft. What is the surface area you would need to paint?

2. EARTH The Equator and the Prime Meridian are perpendicular great circles that divide Earth into North, South and East, West hemispheres. If Earth has a surface area of 197,000,000 square miles, what is the surface area of the North-East section of Earth?

Source: NASA

3. OCEAN If the oceans cover 70% of Earths surface, what is the surface area of the oceans?

Source: NASA

4. GEOMETRY Three nonidentical lines on the circle divide it into either 6 sections or 8 triangles. What condition is needed so that the three lines form 6 sections?

5. GEOGRAPHY Latitude and longitude lines are imaginary lines on Earth. The lines of latitude are horizontal concentric circles that help to define the distance a place is from the equator. Lines of latitude are measured in degrees. The equator is 0. The north pole is 90 north latitude. The lines of longitude are great circles that help to define the distance a place is from the Prime Meridan, which is located in England and considered the longitude of 0.

a. The mean radius of Earth is 3963 miles. Atlanta, Georgia, has coordinates (33N, 84W) and Cincinnati, Ohio, has coordinates (39N, 84W). Estimate the distance between Atlanta and Cincinnati to the nearest tenth.

b. Seattle, Washington, has coordinates (47N, 122W) and Portland, Oregon,has coordinates (45N, 122W). Estimate the distance between Portland and Seattle to the nearest tenth.

A

C

D B


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


12-7

ProjectionsWhen making maps of Earth, cartographers must show a sphere on a plane. To do this they have to use projections, a method of converting a sphere into a plane. But these projections have their limitations.

The map on the right is a Mercator projection of Earth. On this map Greenland appears to be the same size as Africa. But Greenland has a land area of 2,166,086 square kilometers and Africa has a land area of 30,365,700 square kilometers.

The map on the right is a Lambert projection. When a pilot draws a straight line between two points on this map the line shows true bearing, or relative direction to the North Pole. However, the bottom area of this map distorts distances.

1. When would it be useful to use a Mercator projection of Earth?

2. Does each square on the Mercator projection have the same surface area? Explain.

3. Does each square on the Lambert projection have the same surface area? Explain.

4. The Mercator projection uses a cylinder to map Earth, while the Lambert projection uses a cone to map Earth. What other shapes do you think could be used to map Earth?

EnrichmentSpherical Geometry

60 N

180 W 180 E90 W 0 90 E

40 N20 N020 S40 S60 S


Less

on

12-

8

Co

pyr

ight

G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

NAME DATE PERIOD

PDF Pass


Identify Congruent or Similar Solids Similar solids have exactly the same shape but not necessarily the same size. Two solids are similar if they are the same shape and the ratios of their corresponding linear measures are equal. All spheres are similar and all cubes are similar. Congruent solids have exactly the same shape and the same size. Congruent solids are similar solids with a scale factor of 1:1. Congruent solids have the following characteristics:

Corresponding angles are congruent Corresponding edges are congruent Corresponding faces are congruent Volumes are equal

Determine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

ratio of width: 3 6 = 1

2 ratio of length: 4

8 = 1

2

ratio of hypotenuse: 5 10

= 1 2 ratio of height: 4

8 = 1

2

The ratios of the corresponding sides are equal, so the triangular prisms are similar. The scale factor is 1:2. Since the scale factor is not 1:1, the solids are not congruent.

ExercisesDetermine whether the pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor.

1.

2 cm

1 cm

10 cm

5 cm

2.

4.2 in. 12.3 in.

4.2 in.

12.3 in.

3.

4 in.

8 in.

4.

2 m

2 m

4 m1 m

1 m3 m

Study Guide and InterventionCongruent and Similar Solids

12-8

5 in.

4 in.

4 in.3 in.

10 in.

8 in.

8 in.

6 in.Example


Co

pyrig

ht G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


Properties of Congruent or Similar Solids When pairs of solids are congruent or similar, certain properties are known.

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

Two spheres have radii of 2 feet and 6 feet. What is the ratio of the volume of the small sphere to the volume of the large sphere?

Fi

Chapter 12 Resource Masters - Ms....

Documents

Transcript of Chapter 12 Resource Masters - Ms....