Similar Shapes and Scale Drawings

Name ________________________________________ Date __________________ Class __________________

164

Practice and Problem Solving: A/B

1. The plan of a terrace is shown at right. The scale is

2 inches : 6 feet. What are the length and width of

the terrace? Find the terrace’s area.

Length:

___________________________

Width:

___________________________

Area:

___________________________

2. The floor plan of a ballroom is shown at right. The scale is

3 centimeters : 4 meters. What are the length and width of the

ballroom? Find the ballroom’s area.

Length:

___________________________

Width:

___________________________

Area:

___________________________

3. A garage floor measures 150 feet by 120 feet. A scale drawing

of the floor on grid paper uses a scale of 1 unit : 15 feet. What

are the dimensions of the drawing?

_________________________________________________________________________________________

4. The scale model of a skyscraper being built is 4.2 feet tall.

a. When it is finished, the skyscraper will be 525 meters tall. What

scale was used to make the model?

_____________________________________________________________________

b. The model is made out of a stack of plywood sheets. Each sheet is

0.6 inch thick. How many sheets of plywood tall is the model?

_____________________________________________________________________________________

5. You have been asked to build a scale model of a restaurant out of

bottle caps. The restaurant is 20 feet tall. Your scale is 2.4 cm : 1 foot.

a. A bottle cap is 1.2 cm tall. About how many bottle caps tall will your

model be?

_____________________________________________________________________________________

b. You are out of bottle caps, and decide to use popsicle sticks

instead. You measure them, and they are 15.2 cm tall. How many

popsicle sticks tall will your model be?

_____________________________________________________________________________________

LESSON

8-1

Name ________________________________________ Date __________________ Class __________________

165

Similar Shapes and Scale Drawings

Practice and Problem Solving: C

1. A scale drawing of a patio is shown at right. The scale is

4 inches : 6.8 feet. What are the length and width of the

patio? Find the patio’s area.

Length:

___________________________

Width:

___________________________

Area:

___________________________

2. The scale for a scale drawing is 8 millimeters : 1 centimeter. Which is

larger, the actual object or the scale drawing? Explain.

_________________________________________________________________________________________

3. Carol has a small copy of a painting. The dimensions of

her copy are shown at right. The scale of the copy is

0.5 inches : 12 centimeter.

a. Find the dimensions of the original painting.

__________________________________________________

b. Find the area of the original painting.

__________________________________________________

c. One centimeter is equal to approximately 0.033 foot. Find the

dimensions of the original painting in feet. Round to the nearest

thousandth.

_____________________________________________________________________________________

d. Find the area of the original painting in square feet. Round to the

nearest thousandth.

_____________________________________________________________________________________

4. On a scale drawing, a bookshelf is 8 inches tall. The scale factor is

1

8

.

What is the height of the bookshelf?

_________________________________________________________________________________________

5. On a scale drawing, a school is 1.6 feet tall. The scale factor is

1

22

.

Find the height of the school.

_________________________________________________________________________________________

LESSON

8-1

Name ________________________________________ Date __________________ Class __________________

166

Similar Shapes and Scale Drawings

Practice and Problem Solving: D

1. A room in a house is shown on a blueprint. The blueprint has a scale

of 5 inches : 8 feet. A wall in the same blueprint is 30 inches. Complete

the table. The first column is done for you.

Blueprint Length (in.)

5 10 20

Actual Length (ft)

8

16

24 40 48

a. How long is the actual wall? ______________________________________

b. A door in the room has a width of 4 feet. What is the width of the

door in the blueprint?

_____________________________________________________________________________________

2. The scale of a room in a blueprint is 2 inches : 1 foot. A window in the

same blueprint is 12 inches. Complete the table.

Blueprint Length (in.)

2 4 8 12

Actual Length (ft)

1 3 5

a. How long is the actual window? ___________________________________

b. A mantel in the room has an actual width of 8 feet. What is the

width of the mantel in the blueprint?

_____________________________________________________________________________________

3. The scale in the drawing is 2 inches : 4 feet. What are the

length and width of the actual room? Find the area of the

actual room.

Length:

___________________________

Width:

___________________________

Area:

___________________________

4. A studio apartment has a floor that measures 80 feet by 64 feet.

A scale drawing of the floor on grid paper uses a scale of

1 unit : 8 feet. What are the dimensions of the scale drawing?

_________________________________________________________________________________________

LESSON

8-1

Name ________________________________________ Date __________________ Class __________________

167

Similar Shapes and Scale Drawings

Reteach

The dimensions of a scale model or scale drawing are related to the

actual dimensions by a scale factor. The scale factor is a ratio.

The length of a model car is 9 in.

The length of the actual car is 162 in.

9

162

can be simplified to

1

18

.

If you know the scale factor, you can use a proportion to find the

dimensions of an actual object or of a scale model or drawing.

• The scale factor of a model train set is

1

87

. A piece of track in the

model train set is 8 in. long. What is the actual length of the track?

model length

881

696

actual length 87

x

xx

===

The actual length of track is 696 inches.

• The distance between 2 cities on a map is 4.5 centimeters. The map

scale is 1 cm : 40 mi.

4.5 cm 1 cm

distance on map

actual distance mi 40 mi

180

x

==

=

The actual distance is 180 miles.

Identify the scale factor.

1. Photograph: height 3 in. 2. Butterfly: wingspan 20 cm

Painting: height 24 in. Silk butterfly: wingspan 4 cm

photo height in.

painting height in.

==

silk butterfly cm

butterfly cm

==

Solve.

3. On a scale drawing, the scale factor

is

1

12

. A plum tree is 7 inches tall on the

scale drawing. What is the actual height

of the tree?

4. On a road map, the distance between

2 cities is 2.5 inches. The map scale

is 1 inch:30 miles. What is the actual

distance between the cities?

________________________________________ ________________________________________

LESSON

8-1

9in.

99 1

162 in. 162 9 18

÷

==

÷

The scale factor is

1

18

.

Name ________________________________________ Date __________________ Class __________________

168

Similar Shapes and Scale Drawings

Reading Strategies: Read a Map

A scale drawing has the same shape, but is not the same size,

as the object it represents. A map is an example of a scale drawing.

This is a map of a campground. The scale is 1 cm:10 ft.

To find how far the campground entrance

is from the canoe rental office, follow the

steps. Use a centimeter ruler to measure.

Step 1: Measure the distance in centimeters.

The distance is 3 centimeters.

Step 2: Set up a proportion using the map scale as one ratio.

1cm 3 cm

10 ft ftx

=

Step 3: Use the proportion. x = 3 • 10

Step 4: Solve to find the value of x. 30x =

The campground entrance is 30 feet from the canoe

rental office.

Use the map to answer each question.

1. How many centimeters is Campsite 3 from the water?

_________________________________________________________________________________________

2. Write a proportion to find the distance from Campsite 3 to the water.

_________________________________________________________________________________________

3. How many centimeters is Campsite 3 from the canoe rental office?

_________________________________________________________________________________________

4. Write a proportion to find the distance from Campsite 3 to the

canoe rental office

.

_________________________________________________________________________________________

LESSON

8-1

Name ________________________________________ Date __________________ Class __________________

169

Similar Shapes and Scale Drawings

Success for English Learners

Problem 1

What is the scale factor?

Model length Model height

11 1 5.5 1

;

A

ctual length 132 12 Actual height 66 12

1

Scale factor .

12

== ==

=

Problem 2

This is a photo of a painting. If you measure the photo, you could find the

measurements of the actual painting.

1

Scale factor is .

15

photo

1

15 painting

→

Write 2 proportions, one for the length and one for the width.

photo length photo width

11

painting length 15 painting width 15

==

1. In Problem 1, what would happen if you used a different scale factor

for the length than you did for the width?

_________________________________________________________________________________________

2. Explain why it is important for the photo and the painting to be in

proportion in Problem 2.

_________________________________________________________________________________________

LESSON

8-1

Name ________________________________________ Date __________________ Class __________________

170

Geometric Drawings

Practice and Problem Solving: A/B

Use each set of line segments to sketch a triangle. If a triangle cannot

be drawn, explain why.

1. 2.

Sketch: Sketch:

________________________________________ ________________________________________

Can each set of line segments form a triangle? Why or why not?

3.

AB =

1

2

mile 4. DE = 0.205 kilometer

BC =

1

3

mile

EF = 0.01 kilometer

AC =

1

4

mile

DF = 0.02 kilometer

________________________________________ ________________________________________

How many triangles are formed by the angles and sides—unique

triangle, no triangle, or many triangles?

5. 6.

________________________________________ ________________________________________

LESSON

8-2

Name ________________________________________ Date __________________ Class __________________

171

Geometric Drawings

Practice and Problem Solving: C

Astronomers often use triangles to compute the distance between

Earth and other planets in the solar system. The diagram below

shows how triangles are used in this process.

1. Triangles ABC and ABD are right triangles. Where are the right angles?

_________________________________________________________________________________________

2. What is the distance AC in this diagram called? (Hint: What part of the

circle is it?) How does it compare to AD?

_________________________________________________________________________________________

3. Without using exact numbers, compare the distance AC with the distance BC?

_________________________________________________________________________________________

4. Without using exact numbers, compare the distances AB and BC.

_________________________________________________________________________________________

5. Since ABC is a right triangle, how does the distance AB compare to

BC? Is it equal, greater than, or less than? Why?

_________________________________________________________________________________________

6. If you did not know that ABC is a right triangle, what kind of triangle would Exercise 4

suggest that ABC is? Why?

_________________________________________________________________________________________

7. The astronomer would like to know the distance AB from Earth’s center to the planet. The

astronomer knows the distance AC and that ABC is a right triangle. What other angles or

sides could the astronomer measure to help find the distance AB?

_________________________________________________________________________________________

LESSON

8-2

Name ________________________________________ Date __________________ Class __________________

172

Geometric Drawings

Practice and Problem Solving: D

Answer the questions. The first one is started for you.

1. 2.

How long are the two line segments? How long are the two line segments?

________________________________________ ________________________________________

To form a triangle, how long must To form a triangle, how long must

a third line segment be? a third line segment be?

________________________________________ ________________________________________

Choose a third side length, and then Choose a third side length, and then

draw the triangle on the grid. draw the triangle on the grid.

Third side length: ______________ Third side length: ______________

Answer the questions. The first one is started for you.

3. 4.

What is the sum of the two angle What is the sum of the two angle

measures? measures?

________________________________________ ________________________________________

If these angles formed a triangle, what If these angles formed a triangle, what

would be the measure of the third angle? would be the measure of the third angle?

________________________________________ ________________________________________

LESSON

8-2

3 units and 4 units

less than 7 units but greater

than 1 unit

101°

Name ________________________________________ Date __________________ Class __________________

173

Geometric Drawings

Reteach

In this lesson, you learned two different sets of conditions for drawing a triangle.

Three Sides

Can these three sides form a triangle?

The condition that a triangle can be formed is

based on this fact:

The sum of the lengths of two shorter

sides is greater than the length of the

longest side.

What are the lengths of the shorter sides?

4 and 5 units

What is the length of the longest side?

8 units

Is 4 + 5 > 8? Yes.

Two Angles and a Side

Why is a common, or included, side

needed? Do these angles and side form

a triangle?

The condition that a triangle can be formed

is based on this fact:

The sum of the measures of the angles in

a plane triangle is 180 degrees.

What would be the measure of the third

angle in a triangle formed from these parts?

180° = 53° + 34° + x°

x° = 180° − 87°

x = 93°

A triangle can be formed, with the angles

53° and 93° having the 5-meter side in

common.

Answer the questions about triangle drawings.

1. Can a triangle be formed with three sides of equal length? Explain

using the model above.

_________________________________________________________________________________________

2. Can a triangle be formed with angles having measures of 30°, 70°, and 110°? Explain

using the model above.

_________________________________________________________________________________________

LESSON

8-2

Name ________________________________________ Date __________________ Class __________________

174

Geometric Drawings

Reading Strategies: Using Graphic Aids

Real-world problems can be solved more easily in some cases when you

draw a diagram to represent the situation.

Example

Two roads meet at a right angle. One road is 5 kilometers in length, and

the other is 8 kilometers in length. A third road is constructed connecting

the ends of the other two roads. What is its minimum length?

Solution

First sketch the situation.

The two sides, AB, and BC, are the legs of a right triangle. A straight line

connecting points A and C would form the hypotenuse. It would also be the

shortest distance between points A and B.

“Minimum length” and “shortest distance” mean the same thing here.

Since points A, B, and C form a right triangle, the Pythagorean Theorem

can be used to find the length of line segment AC.

AC

2

= AB

2

+ BC

2

, so AC

2

= 5

2

+ 8

2

= 25 + 64 = 89;

2

AC = AC = 89 .

So, the length of the road represented by

AC is 89 kilometers.

Check

Use this result to check that three sides form a triangle. The long

side,

89 , is between 81 , or 9, and 100 , or 10. Add 5 and 8 to get 13.

Since 13 > 9 and 13 > 10, the sides form a triangle.

Use another sheet of paper to answer the questions.

1. A carpenter has a 10-foot board and two 4-foot boards. Can these

form a triangular wooden brace? Explain.

2. A father is giving his child a 30-inch long softball bat for her birthday.

He has a rectangular box that has the dimensions of 5 inches by

6 inches by 25 inches. Will the bat fit in the box? Explain.

LESSON

8-2

Name ________________________________________ Date __________________ Class __________________

175

Geometric Drawings

Success for English Learners

Problem 1

Problem 2

Describe the triangle formed by these parts and any others you can compute or measure.

180° = 56° + 34° + 90°

How long are the sides?

3.6 m, 6 m, and 7 m.

Answer the questions.

1. In Problem 1, how would using a compass and a ruler make drawing

the triangle easier?

_________________________________________________________________________________________

2. Is the triangle in Problem 2 a right triangle? How do you know?

_________________________________________________________________________________________

LESSON

8-2

Can these three sides

form a triangle?

Yes, they can.

If you place the two shorter sides end to

end, they are longer than the third side.

So, the three sides can form a triangle.

Name ________________________________________ Date __________________ Class __________________

176

Cross Sections

Practice and Problem Solving: A/B

What is the common set of points for these figures called—an intersection or a cross

section? Place a check mark by the correct name. Describe the geometric figure formed by

the common points. Assume that the two figures have more than one point in common.

1. A circle and the lateral surface of a cone.

Cross section ____________ Intersection ____________

Figure formed: ____________________________________________________________

2. The edge of a square and the base of a pyramid.

Cross section ____________ Intersection ____________

Figure formed: ____________________________________________________________

3. A plane that is perpendicular to the base of a cube and slices through the cube.

Cross section ____________ Intersection ____________

Figure formed: ____________________________________________________________

4. A circle with an area bigger than the base of a pyramid and slicing parallel to the base

through the pyramid between its apex and its base.

Cross section ____________ Intersection ____________

Figure formed: ____________________________________________________________

Name or describe the geometric figure that is shaded. Each shaded

region results from a plane passing through the solid.

5. 6.

________________________________________ ________________________________________

7. 8.

________________________________________ ________________________________________

LESSON

8-3

Name ________________________________________ Date __________________ Class __________________

177

Cross Sections

Practice and Problem Solving: C

Two identical cubes are intersected by a plane, resulting in two different cross sections

as shown. Answer the questions about the two cross sections.

Cross section X Cross section Y

1. The plane that created cross section X sliced through the cube parallel

to one of its sides. What type of figure is this? Explain.

_________________________________________________________________________________________

2. The plane that created cross section Y sliced through the corner of the

cube in such a way that it intersected the midpoints of the sides of the

cube. What type of triangle is formed by this cross section? Explain.

_________________________________________________________________________________________

A plane intersects a cylinder. In Diagram A, the plane is perpendicular to

the long axis of the cylinder. In Diagram B, the plane is tilted relative to the

cylinder axis. In Diagram C, the plane is tilted even more. In Diagram D, the

plane passes through the cylinder along its axis. Answer the questions.

Diagram A Diagram B Diagram C Diagram D

3. What shape are the cross sections in Diagrams A, B, C and D?

_________________________________________________________________________________________

4. Compare the areas of the four cross sections.

_________________________________________________________________________________________

LESSON

8-3

Name ________________________________________ Date __________________ Class __________________

178

Cross Sections

Practice and Problem Solving: D

Describe the shape that is formed by the cross section. The first one

is done for you.

1. 2.

________________________________________ ________________________________________

3. 4.

________________________________________ ________________________________________

Draw cross sections for these figures that are parallel to their bases.

5. 6.

________________________________________ ________________________________________

LESSON

8-3

a triangle that is similar to the base

Name ________________________________________ Date __________________ Class __________________

179

Cross Sections

Reteach

Cross sections can take a variety of shapes, but they are generally related to the

parts of the figures from which they are formed. The angle at which the

intersecting plane “cuts” the figure is also a factor in determining the shape of the

cross section. However, the cross section is always defined as a plane figure in

the situations presented here.

Example 1

When the intersecting plane is parallel to the

base(s) of the figure, the cross section is

often related to the shape of the base. In this

cylinder, the cross section is congruent to the

bases.

What is the shape of the cross section?

The cross section is a circle that is congruent

to each of the bases of the cylinder.

Example 2

When the intersecting plane is

perpendicular to the base(s) of the figure,

the cross section is not always the same

shape as the base. In this cylinder, the

cross section is a rectangle, not a circle.

What is the cross section?

A rectangle having a length equal to the

height of the cylinder and a width equal to

the diameter of the cylinder.

For each solid, draw at least two cross sections with two different

shapes. Describe the cross sections.

1. 2.

________________________________________ ________________________________________

LESSON

8-3

Name ________________________________________ Date __________________ Class __________________

180

Cross Sections

Reading Strategies: Use a Table

Cross sections are often related to parts of the solids from which they are

formed. This makes it possible to organize into a table, information about

each solid, its cross section, and the plane that “cut” the figures to form the

cross section.

Draw each cross section as described. Complete the table. Be as

specific in your descriptions as possible.

Solid

Intersection

Direction

Cross Section

Shape

How Cross Section

Shape Relates

to the Solid

1.

parallel to the

rectangular base

2.

similar to base

3.

4.

a regular pentagon

5.

6.

passing through the

sphere but not

through its center

7.

8.

9.

passing through the

cone’s vertex and

bisecting its base

10.

11.

LESSON

8-3

Name ________________________________________ Date __________________ Class __________________

181

Cross Sections

Success for English Learners

Problem 1

How can you pass a plane through the pyramid so that the cross section is

a rectangle?

Problem 2

Cross sections of a cone and a pyramid are shown below.

How are the two cross sections different?

Cone Pyramid

Cross section: circle Cross section: triangle

Answer the questions.

1. In Problem 1, if the cross section has 4 sides, and one side lies in the base of the pyramid,

what shape is the cross section? Explain.

_________________________________________________________________________________________

2. In Problem 2, how are the cross sections of the cone and pyramid alike?

_________________________________________________________________________________________

LESSON

8-3

Any plane parallel to the base

of the pyramid will make a

cross section that is a

rectangle.

Name the shapes

formed by the

cross sections.

Name ________________________________________ Date __________________ Class __________________

182

Angle Relationships

Practice and Problem Solving: A/B

For Exercises 1–3, use the figure.

1. Name a pair of vertical angles.

_________________________________________________________________________________________

2. Name a pair of complementary angles.

_________________________________________________________________________________________

3. Name a pair of supplementary angles.

_________________________________________________________________________________________

Use the diagram to find each angle measure.

4. If m∠1 = 120°, find m∠3. 5. If m∠2 = 13°, find m∠4.

________________________________________ ________________________________________

6. If m∠3 = 110°, find m∠2. 7. If m∠4 = 65°, find m∠1.

________________________________________ ________________________________________

Find the value of x in each figure.

8. 9.

________________________________________ ________________________________________

10.

11.

________________________________________ ________________________________________

LESSON

8-4

Name ________________________________________ Date __________________ Class __________________

183

Angle Relationships

Practice and Problem Solving: C

For Exercises 1–4, use the figure.

1. Find m∠FCG. 2. Find m∠BCD.

________________________________________ ________________________________________

3. Find m∠FCB. 4. Find m∠ACG.

________________________________________ ________________________________________

Find the value of x in each figure.

5.

6.

________________________________________ ________________________________________

7. 8.

________________________________________ ________________________________________

9. The walking paths at a park meet each

other as shown. What is the measure

of the angle between the Second Path

and Third Path?

______________________________________

10. Three streets meet each other as shown.

What is the measure of the angle between

Oak Street and Elm Street?

______________________________________

LESSON

8-4

Name ________________________________________ Date __________________ Class __________________

184

Angle Relationships

Practice and Problem Solving: D

For Exercises 1–3, use the figure. The first one is done for you.

1. Name a pair of vertical angles.

_________________________________________________________________________________________

2. Name a pair of complementary angles.

_________________________________________________________________________________________

3. Name a pair of supplementary angles.

_________________________________________________________________________________________

Use the diagram to find each angle measure.

The first one is done for you.

4. If m∠3 = 60°, find m∠1. 5. If m∠4 = 100°, find m∠2.

________________________________________ ________________________________________

6. If m∠1 = 50°, find m∠2. 7. If m∠2 = 125°, find m∠3.

________________________________________ ________________________________________

Find the value of x in each figure. The first one is done for you.

8.

9.

________________________________________ ________________________________________

10.

11.

________________________________________ ________________________________________

LESSON

8-4

∠PSQ and ∠MSN

∠1 and ∠3 are vertical angles,

so m∠1 = 60°

m∠ABC + m∠CBD = 90, so x = 30

Name ________________________________________ Date __________________ Class __________________

185

Angle Relationships

Reteach

Complementary Angles Supplementary Angles Vertical Angles

Two angles

whose measures

have a sum of 90°.

Two angles

whose measures have a

sum of 180°.

Intersecting lines

form two pairs

of vertical angles.

Use the diagram to complete the following.

1. Since ∠AQC and ∠DQB

are formed by intersecting lines,

AQB

HJJJJG

and CQD

HJJJJG

, they are:

________________________________________

2. The sum of the measures

of ∠

AQV and ∠VQT is: ___________

So, these angles are:

________________________________________

3. The sum of the measures of ∠AQC and ∠CQB is: ___________

So, these angles are: ________________________________

Find the value of x in each figure.

4.

5.

________________________________________ ________________________________________

6.

7.

________________________________________ ________________________________________

LESSON

8-4

Name ________________________________________ Date __________________ Class __________________

186

Angle Relationships

Reading Strategies: Understanding Vocabulary

You can use definitions of angles to determine the measures of missing

angles.

Pairs of angles can be classified into different types.

Complementary

angles

Supplementary

angles

Adjacent angles Vertical angles

Congruent

angles

The angle

measures equal

90°.

The angle

measures equal

180°.

The angles share

a common vertex

and side.

The angles are

nonadjacent

angles formed by

two intersecting

lines.

The angles

have the

same

measure.

Find m∠DFE.

∠DFE and ∠CFD are supplementary angles. So the angle

measures equal 180°. You can write an equation to find m∠DFE.

m∠DFE + m∠CFD = 180° Original equation

m∠DFE + 38° = 180° Substitute 38° for m∠CFD.

m∠DFE + 38° − 38° = 180° − 38° Subtract 38° from both sides.

m∠DFE = 142° Simplify.

Use the diagram to find each angle measure. Show your work.

1. Find m∠GLH.

_____________________________________________________

2. Find m∠KLJ.

_____________________________________________________

3. Find m∠HLI.

_____________________________________________________

4. Find m∠ILK.

_____________________________________________________

LESSON

8-4

Name ________________________________________ Date __________________ Class __________________

187

Angle Relationships

Success for English Learners

Problem 1

Vertical angles

Complementary angles.

Supplementary angles.

Problem 2

What is m∠x?

1. What is the sum of complementary angles? Supplementary angles?

_________________________________________________________________________________________

2. What is the sum of the measures of the angles in any triangle?

_________________________________________________________________________________________

LESSON

8-4

∠1 and ∠3 are vertical angles.

∠2 and ∠4 are vertical angles.

Vertical angles have the same measure.

∠5 and ∠6 are complementary angles.

Complementary angles have a sum of 90°.

∠7 and ∠8 are supplementary angles.

Supplementary angles have a sum of 180°.

25° + 37° + x = 180°

62° + x = 180°

−62°

x = 118°

Name ________________________________________ Date __________________ Class __________________

188

Modeling Geometric Figures

Challenge

A city has built a combination diving and swimming pool for an

international swimming competition. The diagram shows the top and

side views of the pool. The dimensions are in feet.

1. What shape is the solid represented by

Region A? Find its volume in cubic feet.

__________________________________

2. What shape is the Region B water

surface? Find its area. (Hint: Its area is

12

1

2

()hb b+ .)

__________________________________

3. To find the volume of Region B, start by finding the volume of the “slab”

of water that is “x” feet deep and in the shape found in Exercise 2.

_________________________________________________________________________________________

4. Next, notice that the volume of Region B below the “slab” of water in Exercise

3 is a “half” prism. What is the height of that prism? Find its volume.

_________________________________________________________________________________________

5. Now, find the total volume of Region B by adding the volumes from Exercise 3 and 4.

_________________________________________________________________________________________

6. What shape is the diving well, Region C? (Hint: Its volume is

3

4

.

3

r

π

)

What part of the volume of the figure is represented by Region C?

Find that volume.

_________________________________________________________________________________________

7. Find the total volume of the swimming pool, Regions A, B, and C.

_________________________________________________________________________________________

8. What value of x would make the total volume of the pool 33,000 cubic feet?

_________________________________________________________________________________________

MODULE

8

Name ________________________________________ Date __________________ Class __________________

189

Circumference

Practice and Problem Solving: A/B

Find the circumference of each circle. Use 3.14 or

22

7

for

π

. Round to

the nearest hundredth, if necessary.

1. 2. 3.

________________________ _______________________ ________________________

4. 5. 6.

________________________ _______________________ ________________________

7. 8. 9.

________________________ _______________________ ________________________

Solve.

10. A circular swimming pool is 21 feet in diameter. What is the

circumference of the swimming pool? Use

22

7

for

π

.

_________________________________________________________________________________________

11. A jar lid has a diameter of 42 millimeters. What is the circumference of

the lid? Use

22

7

for

π

.

_________________________________________________________________________________________

12. A frying pan has a radius of 14 centimeters. What is the circumference

of the frying pan? Use

22

7

for

π

.

_________________________________________________________________________________________

LESSON

9-1

Name ________________________________________ Date __________________ Class __________________

190

Circumference

Practice and Problem Solving: C

Find the circumference of each circle. Use 3.14 or

22

7

for

π

. Round to

the nearest hundredth, if necessary.

1. 2. 3.

________________________ _______________________ ________________________

4. 5. 6.

________________________ _______________________ ________________________

Solve each problem.

7. The circumference of a clock is 22 inches. What is the radius of

the clock?

_________________________________________________________________________________________

8. The circumference of a circular hot tub at a hotel is 56.5 yards.

What is the diameter of the hot tub?

_________________________________________________________________________________________

9. In NCAA basketball rules, the basketball can have a maximum

circumference of 30 inches. What is the maximum diameter of an

NCAA basketball to the nearest hundredth?

_________________________________________________________________________________________

10. Melanie wants to put ribbon around the circumference of a circular

section of the city park. Ribbon comes in rolls of 40 feet. The radius of

the section of the park is 100 feet. How many rolls of ribbon should

Melanie buy?

_________________________________________________________________________________________

LESSON

9-1

Name ________________________________________ Date __________________ Class __________________

191

Circumference

Practice and Problem Solving: D

Find the circumference of each circle. Use 3.14 or

22

7

for

π

. Round to

the nearest tenth, if necessary. The first one is done for you.

1. 2.

________________________________________ ________________________________________

3. 4.

________________________________________ ________________________________________

5. 6.

________________________________________ ________________________________________

Solve each problem.

7. A circular patio has a diameter of 35 yards. What is the circumference

of the patio? Use

22

7

for

π

.

_________________________________________________________________________________________

8. A paper plate has a diameter of 9 inches. What is the circumference of

the plate? Use

22

7

for

π

.

_________________________________________________________________________________________

9. A circular light fixture has a radius of 20 centimeters. What is the

circumference of the light fixture? Use

22

7

for

π

.

_________________________________________________________________________________________

LESSON

9-1

C = 2

π

r ≈ 2(3.14)(8) ≈ 50.24; 50.2 m

Name ________________________________________ Date __________________ Class __________________

192

Circumference

Reteach

The distance around a circle is called the circumference. To find

the circumference of a circle, you need to know the diameter or the

radius of the circle.

The ratio of the circumference of any circle to its diameter

C

d

⎛⎞

⎜⎟

⎝⎠

is always the same. This ratio is known as

π

(pi) and has a value

of approximately 3.14.

To find the circumference C of a circle if you know the diameter

d, multiply

π

times the diameter. C =

π

• d, or C ≈ 3.14 • d.

C =

π

• d

C ≈ 3.14 • d

C ≈ 3.14 • 6

C ≈ 18.84

The circumference is about 18.8 in.

to the nearest tenth.

The diameter of a circle is twice as long as the radius r, or d = 2r.

To find the circumference if you know the radius, replace d with 2r in

the formula. C =

π

• d =

π

• 2r

Find the circumference given the

diameter.

1. d = 9 cm

C =

π

• d

C ≈ 3.14 • ________

C ≈ ___________

The circumference is ________ cm to

the nearest tenth of a centimeter.

Find the circumference given the

radius.

2. r = 13 in.

C =

π

• 2r

C ≈ 3.14 • (2 • ________)

C ≈ 3.14 • ________

C ≈ ___________

The circumference is ________ in. to

the nearest tenth of an inch.

Find the circumference of each circle to the nearest tenth.

Use 3.14 for

π

.

3.

4. 5.

________________________ _______________________ ________________________

LESSON

9-1

Name ________________________________________ Date __________________ Class __________________

193

Circumference

Reading Strategies: Using a Graphic Organizer

Perimeter is the distance around a polygon.

The chart below shows formulas for finding the circumference of circles.

Circumference

of Circle

C =

π

d

(Circumference = pi • diameter)

Circumference

of Circle

C = 2

π

r

(Circumference = 2 • pi • radius)

Use the information in the chart above to complete each exercise.

1. If you knew the radius of a circle, what formula would you use to find

its circumference?

_________________________________________________________________________________________

2. If you knew the diameter of a circle, what formula would you use to find

its circumference?

_________________________________________________________________________________________

3. How does the length of the diameter of a circle relate to the length of

the radius of that same circle?

_________________________________________________________________________________________

4. What values of

π

can you use to approximate the circumference of a

circle?

_________________________________________________________________________________________

5. How does the circumference of a circle relate to the perimeter of a

polygon?

_________________________________________________________________________________________

LESSON

9-1

Measuring the Distance

Around Figures

Name ________________________________________ Date __________________ Class __________________

194

Circumference

Success for English Learners

Problem 1

When you know the length of the radius of a circle,

use the formula C = 2

π

r to find its circumference.

C = 2

π

r

C = 2

π

(6)

C ≈ 2(3.14)(6)

C ≈ 37.68 m

Problem 2

When you know the length of the diameter of a circle,

use the formula C =

π

d to find its circumference.

C =

π

d

C =

π

(11)

C ≈ 3.14(11)

C ≈ 34.54 in.

1. What information do you need to know to use the formula C =

π

d?

_________________________________________________________________________________________

2. A circle has a radius of 9 centimeters. What is the length of its

diameter?

_________________________________________________________________________________________

3. Suppose you know a circle has a diameter of 34 feet. How could you

use the formula C = 2

π

r to find its circumference?

_________________________________________________________________________________________

4. Find the circumference of a circle with a diameter of 10 meters using

both formulas. Show your work.

_________________________________________________________________________________________

LESSON

9-1

Name ________________________________________ Date __________________ Class __________________

195

Area of Circles

Practice and Problem Solving: A/B

Find the area of each circle to the nearest tenth. Use 3.14 for

π

.

1.

A 113 m

2

C 354.9 m

2

B 37.7 m

2

D 452.16 m

2

3. 4. 5.

________________________ _______________________ ________________________

Find the area of each circle in terms of

π

.

6. 7. 8.

________________________ _______________________ ________________________

9. 10. 11.

________________________ _______________________ ________________________

LESSON

9-2

2.

A 201 ft

2

C 25.1 ft

2

B 50.2 ft

2

D 157.8 ft

2

Name ________________________________________ Date __________________ Class __________________

196

Area of Circles

Practice and Problem Solving: C

Find the area of each circle in terms of

π

. Then find the area to the

nearest tenth using 3.14 for

π

.

1. 2. 3.

________________________ _______________________ ________________________

4. 5. 6.

________________________ _______________________ ________________________

Use the formula C

2

= 4

π

A to find the area A of each circle in

terms of

π

.

7. C = 2

π

8. C = 5

π

9. C = 8

π

________________________ _______________________ _________________

Solve.

10. A vanilla cake has a diameter of 8 inches. A chocolate cake has a

diameter of 10 inches. What is the difference in area between the top

surfaces of the two cakes? Use 3.14 for

π

.

_________________________________________________________________________________________

11. What is the difference in area between a circle with a diameter of

3 meters and a square with a side length of 3 meters? Use 3.14 for

π

.

_________________________________________________________________________________________

LESSON

9-2

Name ________________________________________ Date __________________ Class __________________

197

Area of Circles

Practice and Problem Solving: D

Find the area of each circle to the nearest tenth. Use 3.14 for

π

. The

first problem is done for you.

1. 2. 3.

________________________ _______________________ ________________________

4. 5. 6.

________________________ _______________________ ________________________

Find the area of each circle in terms of

π

. The first problem is done for you.

7. 8. 9.

. _______________________ _______________________ ________________________

10. 11. 12.

________________________ _______________________ ________________________

LESSON

9-2

19.6 cm

2

36π cm

2

Name ________________________________________ Date __________________ Class __________________

198

Area of Circles

Reteach

The area of a circle is found by using the formula A =

π

r

2

. To find the area,

first determine the radius. Square the radius and multiply the result by

π

.

This gives you the exact area of the circle.

Example:

Find the area of the circle in terms of

π

.

The diameter is 10 cm. The radius is half the diameter, or 5 cm.

Area is always given in square units.

5

2

= 25

A = 25

π

cm

2

Find the area of each circle in terms of π.

1. A vinyl album with a diameter of 16 inches. 2. A compact disc with a diameter of 120 mm.

_________________ _________________

Sometimes it is more useful to use an estimate of

π

to find your answer.

Use 3.14 as an estimate for

π

.

Example:

Find the area of the circle. Use 3.14 for

π

and round your answer to the

nearest tenth.

The radius is 2.8 cm.

Area is always given in square units.

2.8

2

= 7.84

A = 7.84

π

cm

2

A = 7.84 × 3.14 cm

2

A = 24.6176 cm

2

Rounded to the nearest tenth, the area is 24.6 cm

2

.

Find the area of each circle. Use 3.14 for

π

and round your answer to

the nearest tenth.

3. A pie with a radius of 4.25 inches. 4. A horse ring with a radius of 10 yards.

_________________ _________________

5. A round pond with a diameter of 24 m. 6. A biscuit with a diameter of 9.2 cm.

_________________ _________________

LESSON

9-2

Name ________________________________________ Date __________________ Class __________________

199

Think: Do I know the diameter or the

radius?

The radius goes across half the

circle. The diameter goes across the

whole circle.

Area of Circles

Reading Strategies: Make Connections

Radius Diameter

Find the area of each circle in terms of

π

. Then find the estimated area

using 3.14 for

π

.

The diameter. I can find the radius by

dividing the diameter by 2.

Find the area of each circle in terms of

π

. Then find the estimated area

using 3.14 for

π

.

1.

______________________

LESSON

9-2

2

22

3.14 or

7

Ar

π

=

≈

2

Ar

d

r

d

A

π

=

⎛⎞

=

⎜⎟

⎝⎠

2

8

416

2

Ar

d

r

A

π

ππ

=

⎛⎞

===

⎜⎟

⎝⎠

i

In terms of

π

, the area is 16

π

in.

2

To find the estimated area, use 3.14 for

π

.

2

16

16 3.14

50.24 in

A

π

=

i

2. _________________

_________________

Name ________________________________________ Date __________________ Class __________________

200

Area of Circles

Success for English Learners

2

22

3.14 or

7

Ar

π

=

≈

2

A

r

d

r

d

A

π

=

⎛⎞

=

⎜⎟

⎝⎠

Problem 1

A. The radius is given.

2

3

39

93.14

28.3 m

Arr

A

π

•

= =

==

≈

B. The diameter is given.

2

8

4

22

416

16 3.14

50.2 in.

Ar

d

r

A

π

•

=

===

==

≈

Problem 2

Find the area in terms of .

π

A. The radius is given.

2

7

749

49 ft

Arr

A

π

•

==

=

B. The diameter is given.

2

9

4.5

22

4.5 20.25

20.25 in.

Ar

d

r

A

π

•

=

===

==

=

d = 9 in.

Find the area in terms of

π

. Then use 3.14 for

π

and find the area to the

nearest tenth.

1.

_________________ 2. _________________

_________________ _________________

LESSON

9-2

Name ________________________________________ Date __________________ Class __________________

201

Area of Composite Figures

Practice and Problem Solving: A/B

Estimate the area of each figure. Each square represents

1 square foot.

1. 2.

________________________________________ ________________________________________

Find the area of each figure. Use 3.14 for

π

.

3. 4. 5.

________________________ _______________________ ________________________

6. 7. 8.

________________________ _______________________ ________________________

9. Marci is going to use tile to cover her terrace. How much tile does

she need?

_____________________________________

LESSON

9-3

Name ________________________________________ Date __________________ Class __________________

202

Area of Composite Figures

Practice and Problem Solving: C

Estimate the area of each figure. Each square represents 1 square foot.

1. 2.

________________________________________ ________________________________________

Find the area of each figure. Use 3.14 for

π

.

3. 4. 5.

________________________ _______________________ ________________________

6. The figure shows the dimensions of a room in which

receptions are held. The room is being carpeted.

The three semi-circular parts of the room are

congruent. How much carpet is needed?

_____________________________________

7. A polygon has vertices at F(−5, 2), G(−3, 2), H(−3, 4),

J(1, 4), K(1, 1), L(4, 1), M(4, −2), N(6, −2), P(6, −3), and

Q(−5, −3). Graph the figure on the coordinate plane.

Then find the area and perimeter of the figure.

_____________________________________

LESSON

9-3

Name ________________________________________ Date __________________ Class __________________

203

Area of Composite Figures

Practice and Problem Solving: D

Estimate the area of each figure. Each square represents 1 square

foot. Choose the letter for the best answer. The first one is done

for you.

1. 2.

A 10 ft

2

C 14 ft

2

A 24 ft

2

C 32 ft

2

B 11 ft

2

D 15 ft

2

B 26 ft

2

D 36 ft

2

Find the area of each figure. Use 3.14 for

π

. The first one is done

for you.

3. 4. 5.

________________________ _______________________ ________________________

6. 7. 8.

________________________ _______________________ ________________________

9. The figure shows the dimensions of a room. How much carpet is

needed to cover its floor?

_____________________________________

LESSON

9-3

17 ft

2

Name ________________________________________ Date __________________ Class __________________

204

Area of Composite Figures

Reteach

When an irregular figure is on graph paper, you can estimate

its area by counting whole squares and parts of squares.

Follow these steps.

• Count the number of whole squares. There are

10 whole squares.

• Combine parts of squares to make whole squares

or half-squares.

Section 1 = 1 square

Section 2

≈ 1

1

2

squares

Section 3

1

2

≈ squares

• Add the whole and partial squares

10 + 1 + 1

1

2

+ 1

1

2

= 14

The area is about 14 square units.

Estimate the area of the figure.

1. There are _______ whole squares in the figure.

Section 1 ≈

_______ square(s)

Section 2 =

_______ square(s)

Section 3 =

_______ square(s)

A = _______ + _______ + _______ + _______ = _______ square units

You can break a composite figure into shapes that you know. Then use

those shapes to find the area.

A (rectangle) = 9 × 6 = 54 m

2

A (square) = 3 • 3 = 9 m

2

A (composite figure) = 54 + 9 = 63 m

2

Find the area of the figure.

2. A (rectangle) = _______ ft

2

A (triangle) = _______ ft

2

A (composite figure) = _______ + _______ = _______ ft

2

LESSON

9-3

Name ________________________________________ Date __________________ Class __________________

205

Area of Composite Figures

Reading Strategies: Make Connections

22 2

square 10 100 fts== =

()

2

1

trapezoid 15 10 20 7.5(30) 225 ft

2

=+==i

2

square trapezoid 100 225 325 ft+=+=

Find the area of each figure. Use 3.14 for

π

.

1. 2.

________________________________________ ________________________________________

3. The figure shows the

dimensions of a room. How

much carpet is needed to cover

the floor of the room?

_____________________________________

Shape Area Formula

Triangle

A =

1

2

bh

Square

A = s

2

Rectangle

A =

l

w

Parallelogram

A = bh

Trapezoid

A =

1

2

hb

1

+ b

2

()

Circle

A =

π

r

2

LESSON

9-3

Name ________________________________________ Date __________________ Class __________________

206

Area of Composite Figures

Success for English Learners

Problem 1

Find the area.

Problem 2

Find the area.

Figure 1 = square

Area of square: A = s

2

s = 12

s

2

= 12

2

= 144

A = 144 m

2

Total area ≈ 144 + 56.52 ≈ A ≈ 200.52 m

2

1. How can you find the area of composite figures?

_________________________________________________________________________________________

LESSON

9-3

Step 1: Count the number of whole

squares: 35

Step 2: Count the number of half

squares: 6



Step 3: 6 ÷ 2 = 3



Step 4: 35 + 3 = 38

So, the area is 38 square units.

Figure 2 is a semicircle

Area of semicircle:

(

)

2

1

2

Ar

π

=

Use 3.14 for

π

.

r = 6 so

2

36r =

()

1

3.14 36

2

A

•≈

()

1

113.04

2

A ≈

56.52A

≈

m

2

Substitute.

Multiply.

Multiply by

1

2

.

Name ________________________________________ Date __________________ Class __________________

207

Solving Surface Area Problems

Practice and Problem Solving: A/B

Find the surface area of each solid figure.

1.

_____________________________________ 2. _____________________________________

3.

_____________________________________ 4. _____________________________________

Use the situation below to complete Exercises 5–6.

Cydney built a display stand out of two cubes. The larger cube is

8 inches on each side. The smaller cube is 4 inches on each side.

She painted the display stand after

she put the two cubes

together. She did NOT paint the bottom of the display stand. What

was the total area she painted?

5. Explain your plan for solving the problem.

_________________________________________________________________________________________

6. Solve the problem.

________________________________________

LESSON

9-4

Name ________________________________________ Date __________________ Class __________________

208

Solving Surface Area Problems

Practice and Problem Solving: C

Find the surface area of each solid figure. Round each answer to the nearest tenth.

1.

_____________________________________ 2. _____________________________________

Three students constructed three-dimensional figures of cardboard.

A. B. C.

3. Read each description in Exercises 4–6. Before doing any

calculations, guess which figure has the greatest surface area and

which has the least surface area.

_________________________________________________________________________________________

4. Josh made his figure from 6 congruent squares. The edge of each

square was 8 inches. Which figure did Josh construct? What is the

surface area of his figure?

_________________________________________________________________________________________

5. Kayla used 4 isosceles and one square for her figure. Each edge of

the square was 10 inches. The two congruent sides of the triangles are

13 inches long. The height of the triangles is 12 inches. Which figure

did Kayla construct? What is the surface area of her figure?

_________________________________________________________________________________________

6. Angelica used four congruent equilateral triangles. Each side of the

triangles was 14 inches. The height of each triangle was 12.1 inches.

Which figure did she construct? What is the surface area of her figure?

_________________________________________________________________________________________

7. Check your guesses from Exercise 3. Were you correct? ________________________

LESSON

9-4

Name ________________________________________ Date __________________ Class __________________

209

Solving Surface Area Problems

Practice and Problem Solving: D

Find the surface area of each solid figure. The first one is done for you.

1.

_____________________________________ 2. _____________________________________

3. _____________________________________ 4. _____________________________________

Use the situation below to answer 5–7.

Cydney built a display stand out of two cubes. The larger cube is 8 inches

on each side. The smaller cube is 4 inches on each side. She painted the

entire outside of each cube before she put the cubes together.

5. What was the surface area she

painted for the smaller cube?

___________________________

6. What was the surface area she

painted for the larger cube?

___________________________

7. What was the total area that

she painted on both cubes?

___________________________

LESSON

9-4

286 ft

2

Name ________________________________________ Date __________________ Class __________________

210

Solving Surface Area Problems

Reteach

The surface area of a three-

dimensional figure is the combined

areas of the faces.

You can find the surface area of a

prism by drawing a net of the

flattened figure.

Notice that the top and bottom have

the same shape and size. Both sides

have the same shape and size. The

front and the back have the same

shape and size.

Remember: A = lw

Since you are finding area, the answer will be in square units.

Find the surface area of the prism formed by the net above.

1. Find the area of the front face: A

= ____ • ____ = _________________ in

2

.

The area of the front and back faces is 2 •

____ = _________________ in

2

.

2. Find the area of the side face: A

= ____ • ____ = _________________ in

2

.

The area of the 2 side faces is 2

• ____ = _________________ in

2

.

3. Find the area of the top face: A

= ____ • ____ = _________________ in

2

.

The area of the top and bottom faces is 2 •

____ = _________________ in

2

.

4. Combine the areas of the faces:

____ + ____ + ____ = _________________ in

2

.

5. The surface area of the prism is

_________________ in

2

.

Find the surface area of the prism formed by each net.

6. 7.

. ___________________

LESSON

9-4

Name ________________________________________ Date __________________ Class __________________

211

Solving Surface Area Problems

Reading Strategies: Analyze Information

The surface area of a three-dimensional figure is the total area of

all its surfaces.

If you analyze the net of a rectangular prism, you notice there are

six faces. Each face pairs up with another, congruent face:

To find the surface area of a rectangular prism, find the sum of the areas of

the six faces. Since you are finding area, the answer will be in square units.

Remember: A = lw.

Use the congruent pairs of faces to simplify the computation.

Left side and right side: 2 × (6 × 4) = 48

Back and front: 2 × (8 × 4) = 64

Top and bottom: 2 × (6 × 8) = 96

Sum of areas: 48 + 64 + 96 = 208

Total surface area: 208 square inches

Find the surface area of each rectangular prism.

1.

2.

________________________________________ ________________________________________

LESSON

9-4

Name ________________________________________ Date __________________ Class __________________

212

Solving Surface Area Problems

Success for English Learners

If you unfold a three-dimensional figure and lay it flat, you have made a net.

A net is a two-dimensional shape that lets you picture all the surfaces

of a three-dimensional figure. A net helps you see how much surface a

three-dimensional figure covers.

The surface area of a three-dimensional figure is the total area of all its

surfaces. Surface area is measured in square units. Remember: A = lw

Complete.

1. What is the area of the top?

_____________________________________

2. What is the area of the bottom?

_____________________________________

3. What is the area of the front?

_____________________________________

4. What is the area of the back?

_____________________________________

5. What is the area of the left side?

_____________________________________

6. What is the area of the right side?

_____________________________________

7. What is the total surface area of the figure?

_____________________________________

8. What do you notice about pairs of surfaces of a rectangular prism that

have the same areas?

_________________________________________________________________________________________

LESSON

9-4

Name ________________________________________ Date __________________ Class __________________

213

Solving Volume Problems

Practice and Problem Solving: A/B

Find the volume of each figure.

1.

2.

________________________________________ ________________________________________

3. 4.

________________________________________ ________________________________________

Using cheese, Theo made the display shown at right.

Use the figure to complete Exercises 5–7.

5. How many cubic centimeters of cheese are in the

completed display?

________________________________________

6. Each kilogram of the cheese Theo used takes up a

volume of about 20 cubic centimeters. What is the

approximate mass of Theo’s display?

________________________________________

7. Theo’s friend made a display with dimensions that

were each half as long as those Theo used. What

is the approximate mass of Theo’s friends display?

________________________________________

LESSON

9-5

Name ________________________________________ Date __________________ Class __________________

214

Solving Volume Problems

Practice and Problem Solving: C

Find the volume of each figure.

1.

2.

________________________________________ ________________________________________

3. 4.

________________________________________ ________________________________________

For the school carnival, Pietro built a clear plastic

container to be used in a game called Guess How

Many Marbles. The container is to be filled with

marbles that are 12 millimeters in diameter.

5. Find the volume of the entire container.

________________________________________

6. Marsha says that one marble has a volume of

904.3 cm

2

, so the container can’t hold any

marbles. What error did Marsha make?

_________________________________________________________________________________________

7. Can you find how many marbles fit in the container by dividing the

volume of the container by the volume of 1 marble? Explain your

reasoning.

_________________________________________________________________________________________

LESSON

9-5

Name ________________________________________ Date __________________ Class __________________

215

Solving Volume Problems

Practice and Problem Solving: D

Tell how many cubes are in each figure. The first one is done for you.

1.

2.

________________________________________ ________________________________________

Find the volume of each figure. The first one is done for you.

3. 4.

________________________________________ ________________________________________

For a school project, students have to build a pyramid of cubes.

5. Each cube will be like the one shown at the right.

What are the dimensions of the cube?

length:

________ mm height: ________ mm

width:

________ mm

6. What is the volume of the cube at the right?

________________________________________

The completed pyramid will look like the figure shown at the right.

7. How many cubes are in the pyramid?

________________________________________

8. What is the volume of the entire pyramid?

________________________________________

LESSON

9-5

Think: There are 3 × 2

cubes in each layer.

There are 2 layers. So,

there are 3 × 2 × 2 cubes.

12 cubes

Think: V = lwh

V = 7 × 3 × 5 = 105

105 in

3

Name ________________________________________ Date __________________ Class __________________

216

Solving Volume Problems

Reteach

The volume of a solid figure is the number of cubic units inside the figure.

A prism is a solid figure that has

length, width, and height.

Each small cube represents

one cubic unit.

Volume is measured in cubic units, such as

in

3

, cm

3

, ft

3

, and m

3

.

The volume of a solid figure is the product of

the area of the base (B) and the height (h).

Rectangular Prism

The base is a rectangle.

To find the area of the base,

use B = lw.

Triangular Prism

The base is a triangle.

To find the area of the base,

use B =

1

2

bh.

Trapezoidal Prism

The base is a trapezoid.

To find the area of the base,

use

B =

1

2

(b

1

+ b

2

)h.

Find the volume of each figure.

1.

2.

3.

________________________ _______________________ ________________________

LESSON

9-5

V = Bh

Name ________________________________________ Date __________________ Class __________________

217

Solving Volume Problems

Reading Strategies: Use a Graphic Organizer

You can use a graphic organizer to help when you are solving volume

problems.

Volume of all Prisms

V = Bh

Volume of a Rectangular Prism

The base is a rectangle. To find the area of the

base, use B

= lw.

V = Bh = lwh

Volume of a Triangular Prism

The base is a triangle. To find the area of the

base, use B

=

1

2

bh.

V = Bh =

1

2

bh

b

h

p

Volume of a Trapezoidal Prism

The base is a trapezoid. To find the area of the

base, use

B =

1

2

(b

1

+ b

2

)h.

V = Bh =

1

2

(b

1

+ b

2

)h

b

h

p

Find the volume of each figure.

1.

2.

3.

________________________ _______________________ ________________________

LESSON

9-5

volume

of a

prism

area of

the base

of prism

height of

the prism

Name ________________________________________ Date __________________ Class __________________

218

Solving Volume Problems

Success for English Learners

The volume of a prism is equal to the area of the base of the prism times the

height of the prism.

Problem 1

Some prisms have more than one measure that is labeled “height.”

Volume of a Triangular Prism

In a triangular prism, there is the height of the

base triangle and the height of the prism.

Sometimes subscripts (small letters to the right

of a letter) can help you know which “height”

to use.

h

b

is the height of the base.

h

p

is the height of the prism.

The base is a triangle. To find the area of the

base, use B =

1

2

bh

b

.

V = Bh =

1

2

bh

b

h

p

Problem 2

Some prisms have more than one measure that is labeled “base.”

Volume of a Trapezoidal Prism

In a trapezoidal prism, the area of the base is

the average of the lengths of the two bases

times the height.

The subscripts tell you to use both bases

(

b

1

and b

2

) when using this formula.

The trapezoidal prism also has two “heights.”

h

b

is the height of the base.

h

p

is the height of the prism.

The base is a trapezoid. To find the area of the

base, use

B =

1

2

(b

1

+ b

2

)h

b

.

V = Bh =

1

2

(b

1

+ b

2

)h

b

h

p

Find the volume of each figure.

1. 2.

________________________________________ ________________________________________

LESSON

9-5

Name ________________________________________ Date __________________ Class __________________

219

Circumference, Area, and Volume

Challenge

The Schultz family is planning a garden for part of their back yard. The width of the garden

will be 20 feet. Each family member has different requirements. The requirements are

shown in the table below.

Name Requirement

Mr. Schultz The maximum length of fencing around the garden is 100 feet.

Mrs. Schultz The area of the garden must be greater than 400 square feet.

Angelica One-half of the garden will be flowers. At most, there will be 350

square feet for flowers.

Robert At least 15% percent of the garden will be used to grow carrots and

lettuce. At the least, there will be 45 square feet for carrots and lettuce.

Write and solve an inequality that represents the possible values for

the widths of the garden for each family member. Then write an

inequality that would satisfy everyone.

1. Mr. Schultz:

___________________________ 2. Mrs. Schultz: ___________________________

3. Angelica:

______________________________ 4. Robert: _________________________________

5. Dimensions for garden that satisfies all 4 people:

width (w):

___________________________ length (l): ___________________________

The family also wants to put a circular fish pond in the backyard. The

pond is to be at least 2 feet deep and have an area of at least 78

4

7

ft

2

and no greater than 154 ft

2

. A 2-foot wide walkway will surround

the pond. The family creates a pond that meets the requirements.

Use

π =

22

.

7

Use that information to answer each question.

6. What are the possible dimensions of the

radius (r) of the pond?

_____________________________

7. What are the possible lengths of the

circumference (C

p

) of the pond? _____________________________

8. What are the possible lengths of the

circumference (C

w

) of the outside of the walkway? _____________________________

9. What are the possible values of the volume (V)

of the pond?

_____________________________

MODULE

9

342

UNIT 4: Geometry

MODULE 8 Modeling Geometric

Figures

LESSON 8-1

Practice and Problem Solving: A/B

1. 15 ft; 6 ft; 90 ft

2

2. 16 m; 12 m; 192 m

2

3. The scale drawing is 10 units by 8 units.

4. a. 1 ft = 125 m

b. 84 sheets of plywood tall

5. a. 40 bottle caps tall

b. approximately 3 popsicle sticks tall

Practice and Problem Solving: C

1. 25.5 ft; 23.8 ft; 606.9 ft

2

2. Because the scale is 8 mm: 1 cm, and

because 1 cm is longer than 8 mm, the

actual object will be larger.

3. a. 42 cm by 126 cm

b. 5,292 cm

2

c. approximately 1.386 ft by 4.158 ft

d. approximately 5.763 ft

2

4. 64 in.

5. 35.2 ft

Practice and Problem Solving: D

1.

Blueprint

length (in.)

5 10

15

20

25 30

Actual

Length (ft)

8 16 24

32

40 48

a. 48 ft

b. 2.5 in.

2.

Blueprint

length (in.)

2 4

6

8

10

12

Actual

Length (ft)

1

2

3

4

5

6

a. 6 ft

b. 16 in.

3. 24 ft; 12 ft; 288 ft

2

4. 10 units by 8 units

Reteach

1. 3 in.; 24 in.;

1

8

2. 4 cm; 20 cm;

1

5

3. 84 in.

4. 75 mi

Reading Strategies

1. 3 cm

2. Sample answer:

13

10

x

=

3. 5 cm

4. Sample answer:

=

15

10

x

Success for English Learners

1. Sample answer: The car would not be in

proportion.

2. Sample answer: If the photo does not

have the same proportions as the

painting, the face will be stretched tall or

stretched wide.

LESSON 8-2

Practice and Problem Solving: A/B

1.

2.

No triangle can be formed because the

sum of the measures of the two shorter

sides has the same measure as the

longest side.

343

3. Yes, because the sum of the measures of

the two shorter sides is greater than the

measure of the longest side,

e.g.,

1

3

+

1

4

>

1

.

2

4. No, because the sum of the measures of

the two shorter sides is less than the

measure of the longest side, e.g., 0.02

+

0.01

< 0.205.

5. Unique; since the sum of the angles is

less than 180

° and a side is included.

6. Many, since the sum of the measures of

the angles is less than 180

° but no side is

included.

Practice and Problem Solving: C

1. They are angles ACB and ADB, formed by

Earth’s radii and the tangent lines running

to the planet.

2. Both are Earth’s radii.

3.

AC is much less than BC.

4.

AB and BC are approximately equal.

5.

AB > BC

6. Isosceles triangle, since

AB and BC are

approximately equal.

7. The astronomer knows that

ACB is a right

angle and the angle

CAB could be

measured. This is enough information to

compute

AB using similar triangles or

trigonometry.

Practice and Problem Solving: D

1. 3 and 4 units; less than 7 units, but

greater than 1 unit; Diagrams will vary.

2. 3 and 7 units; less than 10 units, but

greater than 4 units; Diagrams will vary.

3. 101

°; 79°

4. 129

°; 51°

Reteach

1. Yes; if x is the length of each side, then

x + x > x or 2x > x, so the condition for a

triangle to be formed is met.

2. No. The sum of the measures of the three

angles is greater than 180

°.

Reading Strategies

1. Diagrams may vary, but students should

realize that the two 4-foot boards add up

to 8 feet, which is less than the 10-foot

board, so no triangle can be formed with

the boards.

2. Diagrams and calculations may vary, but

students should first find the hypotenuse

of the right triangle formed by the 5 and

6-inch sides, which is

61

inches. Then,

they should find the length of the

hypotenuse formed by the 25-inch side

and

61

inches, which is

686

inches, or

about 26 inches. A 30-inch bat would not

fit in the box.

Success for English Learners

1. The compass could be used to make two

arcs of radii equal in length to the shorter

segments from each end of the longer

segment. The point of intersection of the

arcs would be where the shorter sides of

the triangle intersect.

2. Yes, the sum of the measures of the

angles given is 90

°, so the third angle has

to be 90 degrees for the sum of the three

angle measures to be 180

°.

LESSON 8-3

Practice and Problem Solving: A/B

1. cross section; The circle is a plane figure

intersecting a three-dimensional curved

surface. The figure formed is a curved line

on the surface of the cone.

2. intersection; The edge of a square is a

straight line and the base of the pyramid is

a plane figure. A straight line is formed.

3. cross section; A square is formed.

4. cross section; The circle is a plane figure.

A polygon results that is similar to the

polygon that forms the base.

5. trapezoid

6. triangle

7. circle

8. ellipse or oval

Practice and Problem Solving: C

1. It is a square. The length of each of its

sides is the same as the length of the side

of the square.

2. An equilateral triangle; Since each of the

segments from the vertex of the cube to

the midpoint of the side is equal and the

344

angles at the vertex are 90º, the third

sides of each triangle are equal and form

the cross section.

3. A: circle; B and C: ellipses or ovals; D: a

plane of length, h, the cylinder’s height,

and width, d, the cylinder’s diameter

4. Area A < Area B < Area C < Area D

Practice and Problem Solving: D

1. a triangle that is similar to the base

2. a rectangle or a square

3. a trapezoid

4. a circle

5. Drawings will vary, but the cross section

should be a regular octagon that is

congruent to the bases of the prism.

6. Drawings will vary, but the cross section

should be a regular pentagon that is

similar to the base of the pyramid.

Reteach

1. Drawings will vary. Sample answers: a

triangular cross section formed by a plane

that is perpendicular to the base of the

pyramid and including its apex point; a

rectangular cross section formed by a

plane that is parallel to the base of the

pyramid

2. Drawings will vary, Sample answers: a

triangular cross section formed by a plane

that is parallel to the prism’s bases and

congruent to them; a rectangular cross

section formed by a plane that is

perpendicular to the bases and having a

length that is equal to the height of the

prism

Reading Strategies

1. Diagrams will vary but should show a

rectangular cross section that is parallel to

the base and similar to it.

2. rectangle

3. Diagrams will vary but should show a

pentagonal cross section that is congruent

to the bases.

4. parallel to the bases

5. congruent to bases

6. Diagrams will vary but should show a

circular cross section of radius less than

the radius of the sphere.

7. circle

8. similar to a circle that is the circumference

of the sphere but smaller than that circle

9. Diagrams will vary but should show a

plane passing through the cone’s vertex,

its lateral surface in two lines, and

bisecting its base.

10. isosceles triangle

11. The two sides of the triangle that are

equal length are the same length as the

slant height of the cone. The third, shorter

side is equal to the diameter of the cone’s

base.

Success for English Learners

1. It is a trapezoid; the edge of the cross

section in the base is longer than and

parallel to the edge of the cross section in

the face of the pyramid.

2. Both cross sections are parallel to the

bases. Each cross section is similar to the

figure’s base.

LESSON 8-4

Practice and Problem Solving: A/B

1. ∠AEB and ∠DEF

2. ∠AEB and ∠BEC

3. Sample answer: ∠AEF and ∠DEF

4. 120°

5. 13°

6. 70°

7. 115°

8. 28

9. 18

10. 22

11. 15

Practice and Problem Solving: C

1. 66°

2. 125°

3. 114°

4. 156°

5. 39

345

6. 43

7. 24

8. 19

9. 41.25°

10. 33°

Practice and Problem Solving: D

1. ∠MSN and ∠PSQ

2. ∠PSQ and ∠QSR

3. Sample answer: ∠MSN and ∠NSP

4. 60°

5. 100°

6. 130°

7. 55°

8. 30

9. 40

10. 35

11. 135

Reteach

1. vertical angles;

2. 90°; complementary angles

3. 180°; supplementary angles

4. 80

5. 20

6. 6

7. 25

Reading Strategies

1. 30°

2. 60°

3. 150°

4. 90°

Success for English Learners

1. 90°; 180°

2. 180°

Module 8 Challenge

1. A rectangular solid; V

A

= 4x(6x)x = 24x

3

2. A trapezoid; A

B

=

12

1

2

()hb b+

=

1

2

4(4 8)

x

xx+ = 24x

2

3. V

B part 1

= A

B

(x) = (24x

2

)x = 24x

3

4. V

B part 2

=

2

11

(3 ) (24 )(3 )

22

B

Ax x x

=

36x

3

5. V

B total

= 24x

3

+ 36x

3

= 60x

3

6. A sphere; one fourth of a sphere;

V

C

=

()

3

14 64

4

43 3

x

π

⎛⎞

=

⎜⎟

⎝⎠

7. V

total

= V

A

+ V

B total

+ V

C

= 24x

3

+ 60x

3

+

3

64

3

x

π

=

3

16

421

3

x

π

⎛⎞

+

⎜⎟

⎝⎠

or approx.

151x

3

.

8. Divide 33,000 by 151 to get about 218.

Take the cube root; x is about 6 feet.

346

MODULE 9 Circumference,

Area, and Volume

LESSON 9-1

Practice and Problem Solving: A/B

1. 18.84 in.

2. 56.52 cm

3. 4.71 ft

4. 25.12 m

5. 37.68 ft

6. 12.56 yd

7. 43.96 in.

8. 26.26 cm

9. 7.85 m

10. 66 ft

11. 132 mm

12. 88 cm

Practice and Problem Solving: C

1. 3.93 in.

2. 11.30 yd

3. 13.19 mm

4. 2.36 cm

5. 4.19 ft

6. 3.14 in.

7. 3.5 in.

8. 18 yd

9. 9.55 in.

10. 16

Practice and Problem Solving: D

1. 50.2 m

2. 62.8 in.

3. 9.4 ft

4. 22.0 mm

5. 18.8 cm

6. 12.6 yd

7. 110 yd

8. 28.3 in.

9. 125.7 cm

Reteach

1. 9; 28.26; 28.3

2. 13; 26; 81.64; 81.6

3. 40.8 cm

4. 31.4 ft

5. 9.4 in.

Reading Strategies

1. C = 2

π

r

2. C =

π

d

3. It is twice as long.

4. Sample answer: 3.14 or

22

7

5. The circumference of a circle is the

distance around a circle. It is given in

units. The perimeter of a polygon is the

distance around a polygon. It is given in

units.

Success for English Learners

1. the length of the diameter.

2. 18 cm

3. Take half of the diameter, 17 ft, and

substitute that value into the formula for

r.

4.

d = 10 so r = 5

C = 2

π

r C =

π

d

= 2 • 3.14 • 5 = 3.14 • 10

= 31.4 = 31.4

347

LESSON 9-2

Practice and Problem Solving: A/B

1. A

2. B

3. 50.2 in.

2

4. 153.9 m

2

5. 254.3 yd

2

6.

π

cm

2

7. 54.76

π

cm

2

8. 25

π

in.

2

9. 121

π

mm

2

10. 6.25

π

ft

2

11. 9

π

m

2

Practice and Problem Solving: C

1. 1.2544

π

cm

2

; 3.9 cm

2

2. 0.0625

π

in.

2

; 0.2 in

2

3. 0.16

π

in.

2

; 0.5 in

2

4. 54.76

π

cm

2

; 171.9 cm

2

5. 36,864

π

yd

2

; 115,753 yd

2

6. 0.49

π

m

2

; 1.5 m

2

7. A =

π

8. A = 6.25

π

9. A = 16

π

10. The area of the 10-inch chocolate cake is

28.26 in

2

larger than the area of the

vanilla cake.

11. The square’s area is 1.935 m

2

larger than

the circle’s area.

Practice and Problem Solving: D

1. 19.6 cm

2

2. 379.9 in.

2

3. 28.3 mm

2

4. 78.5 in

2

5. 132.7 cm

2

6. 162.8 yd

2

7. 36

π

cm

2

8. 90.25

π

in

2

9. 12.25

π

yd

2

10. 121

π

yd

2

11. 9

π

m

2

12. 36

π

ft

2

Reteach

1. 64

π

in

2

2. 3600

π

m

2

3. 56.7 in.

2

4. 314 yd

2

5. 452.2 m

2

6. 66.4 cm

2

Reading Strategies

1. 49

π

cm

2

; 153.86 cm

2

2. 6.25

π

yd

2

; 19.625 yd

2

Success for English Learners

1. 10.24

π

mm

2

; 32.2 mm

2

2. 90.25

π

yd

2

; 283.4 yd

2

LESSON 9-3

Practice and Problem Solving: A/B

Answers may vary for Exercises 1 and 2.

1. 21 ft

2

2. 24 ft

2

3. 90 ft

2

4. 208 m

2

5. 140 ft

2

6. 23.13 m

2

7. 100 ft

2

8. 33.28 m

2

9. 57.12 m

2

348

Practice and Problem Solving: C

Answers may vary for Exercises 1 and 2.

1. 22 ft

2

2. 30 ft

2

3. 104 ft

2

4. 223.4m

2

5. 60.75 m

2

6. 258.39 m

2

7. A = 52 units

2

; P = 36 units

Practice and Problem Solving: D

1. C

2. B

3. 17 ft

2

4. 30.28 m

2

5. 174 ft

2

6. 84 m

2

7. 158.13 ft

2

8. 288 m

2

9. 189.25 ft

2

Reteach

1. 9,

11 11

1, ,1,9,1, ,1,12

22 22

2. 32, 6, 32, 6, 38

Reading Strategies

1. 63 m

2

2. 76 m

2

3. 30.28 m

2

Success for English Learners

1. Separate the figures into simpler figures

whose areas you can find.

LESSON 9-4

Practice and Problem Solving: A/B

1. 142 in

2

2. 190 cm

2

3. 1,236 cm

2

4. 3,380 ft

2

5. Possible answer: I would find the total

surface area of each cube and then

subtract the area of the sides that are not

painted, including the square underneath

the small cube.

6. 384 in

2

Practice and Problem Solving: C

1. 101.4 in

2

2. 797.4 m

2

3. Check students’ guesses.

4. B; 384 in

2

5. C; 340 in

2

6. A; 338.8 in

2

7. Discuss students’ guesses and whether

they were correct or not.

Practice and Problem Solving: D

1. 286 ft

2

2. 1,160 ft

2

3. 80 in

2

4. 124 in

2

5. 96 in

2

6. 384 in

2

7. 480 in

2

349

Reteach

1. 5 • 8 = 40 in

2

; 2 • 40 = 80 in

2

2. 5 • 3 = 15 in

2

;2 • 15 = 30 in

2

3. 3 • 8 = 24 in

2

;2 • 24 = 48 in

2

4. 80 + 30 + 48 = 158 in

2

5. 158 in

2

6. 340 in

2

7. 592 cm

2

Reading Strategies

1. 756 square feet

2. 600 square inches

Success for English Learners

1. 32 cm

2

2. 32 cm

2

3. 8 cm

2

4. 8 cm

2

5. 16 cm

2

6. 16 cm

2

7. 112 cm

2

8. Sample answer: There are 3 pairs of

surfaces with the same areas: the top and

bottom, the left side and right side, the

front and back.

LESSON 9-5

Practice and Problem Solving: A/B

1. 84 in

3

2. 180 cm

3

3. 600 ft

3

4. 360 cm

3

5. 312 cm

3

6. 15.6 kg

7. 1.95 kg

Practice and Problem Solving: C

1. 124.4 in

3

2. 477.8 cm

3

3. 120 m

3

4. 20.2 cm

3

5. 135 cm

3

6. Marsha got the units confused. The

volume of one marble is 7,234.5 mm

3

.

Marsha needs to convert that volume

to cm

3

, which is about 7.2 cm

3

.

7. Answers will vary. Sample answer: If you

divide the volume of the container by the

volume of 1 marble, you can find the

number of marbles that will fit inside the

container. However, the volume of all the

marbles will not equal the volume of the

container, because the marbles are round

and there will be empty space.

Practice and Problem Solving: D

1. 12 cubes

2. 24 cubes

3. 105 in

3

4. 48 m

3

5. length: 10 mm; width: 10 mm;

height: 10 mm

6. 1,000 mm

3

7. 6 cubes

8. 6,000 mm

3

Reteach

1. 80 m

3

2. 120 in

3

3. 72 cm

3

Reading Strategies

1. 60 m

3

2. 720 in

3

3. 108 cm

3

Success for English Learners

1. 216 in

3

2. 108 cm

3

350

MODULE 9 Challenge

1. 2(20 + x) ≤ 100; x ≤ 30

2. 20x > 400; x > 20

3. 0.5(20x) ≤ 350; x ≤ 35

4. 0.15(20x) ≥ 45; x ≥ 15

5. w = 20 feet and 30 feet ≥ l > 20 feet

6. 5 ft ≤ r ≤ 7 ft

7.

3

31

7

ft ≤ C

p

≤ 44 ft

8. 44 ft

≤ C

w

≤

1

56

7

ft

9.

1

157

7

ft

3

≤ V ≤ 308 ft

3