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Similar SolidsAPART B C

Multi-Part Lesson 2

654 Measurement and Proportional Reasoning

Changes in DimensionsMODELS Stephen is creating a model of

the Washington Monument for history class.

The model will be 1 _

100 of the monument’s

actual size.

1. The pyramid that sits atop the

monument’s obelisk shape has

a height of 55.5 feet. What is the

height of the pyramid on the

model Stephen is creating?

2. MAKE A CONJECTURE Write a sentence

about the area of the triangular

side of the model compared

with the actual monument.

Cubes are similar solids because they have the same shape and their

corresponding linear measures are proportional.

The cubes at the right are similar.

The ratio of their corresponding

edge lengths is 8 _ 4 or 2. The scale

factor is 2. How are their surface

areas related?

S.A. of Small Cube S.A. of Large Cube

S.A. = 6(4)(4)

There are 6 faces.

S.A. = 6(2 · 4)(2 · 4)

= 2 · 2(6)(4 · 4)

= 22(6)(4 · 4)

To find the surface area of the large cube, multiply the surface area

of the small cube by the square of the scale factor, 22 or 4. This

relationship is true for any similar solids.

Surface Area of Similar Solids

If Solid X is similar to Solid Y by a scale factor, then the surface area of X is equal to the surface area of Y times the square of the scale factor.

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Main IdeaSolve problems involving similar solids.

Vocabularysimilar solidsVs

4 in. 8 in.

8 in.

8 in.

4 in.

4 in.

Surface Area of Similar Solids

The surface area of a rectangular prism is 78 square centimeters.

What is the surface area of a similar prism that is 3 times as large?

S.A. = 78 × 32 Multiply by the square of the scale factor.

S.A. = 78 × 9 Square 3.

S.A. = 702 cm2 Simplify.

MODELS Refer to the previous page. The surface area of the

exposed portion of the pyramid atop the Washington Monument

is 4,012 square feet. What is the surface area in square inches, to

the nearest tenth, of the pyramid on Stephen’s model?

S.A. = 4,012 × ( 1 _

100 )

2 Multiply by the square of the scale factor.

S.A. = 4,012 × 1 _

10,000 Square 1 _ 100 .

S.A. = 0.4012 ft2 Simplify.

S.A. = 0.4012 ft · ft × 12 in. _

1 ft × 12 in.

_ 1 ft

Convert to inches.

S.A. = 57.8 in2 Simplify.

The surface area of Stephen’s model is 57.8 square inches.

a. The surface area of a triangular prism is 34 square inches. What

is the surface area of a similar prism that is twice as large?

b. RAISINS The world’s largest box of raisins has a surface area of

352 square feet. If a similar box is smaller than the largest box

by a scale factor of 1 _ 48

, what is its surface area?

Refer to the cubes on the previous page.

Volume of Small Cube Volume of Large Cube

V = 4 · 4 · 4 V = (2 · 4)(2 · 4)(2 · 4)

= 2 · 2 · 2(4 · 4 · 4)

= 23(4 · 4 · 4)

The volumes of similar solids are related by the cube of the

scale factor.

Volume of Similar Solids

If Solid X is similar to Solid Y by a scale factor, then the volume of X is equal to the volume of Y times the cube of the scale factor.

Real-World LinkIt takes about 4 1 _ 2 pounds of fresh grapes to make one pound of raisins.

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Lesson 2C Similar Solids 655

656 Measurement and Proportional Reasoning

Volume of Similar Solids

A triangular prism has a volume of 432 cubic yards. If the prism

is reduced to one third its original size, what is the volume of the

new prism?

V = 432 × ( 1 _ 3 )

3 Multiply by the cube of the scale factor.

V = 432 × 1 _ 27

Cube 1 _ 3 .

V = 16 yd3 Simplify.

The volume of the new prism is 16 cubic yards.

c. A square pyramid has a volume of 512 cubic centimeters. What

is the volume of a square pyramid with dimensions one fourth

of the original?

HOCKEY The standard hockey puck

measures as shown at the right. Find

1 in.

1.5 in.

the surface area and volume of the

giant puck at the left. Use 3.14 for π.

Find the volume and surface area

of the standard puck first.

V = πr2h S.A. = 2(πr2) + 2πrh

≈ (3.14)(1.5)2(1) ≈ 2(3.14)(1.5)2 + 2(3.14)(1.5)(1)

≈ 7.065 in3 ≈ 14.13 + 9.42

≈ 23.55 in2

Find the volume and surface area of the giant puck using the

scale factor.

V = V(40)3 S.A. = S.A.(40)2

= (7.065)(40)3 = (23.55)(40)2

= 452,160 in3 = 37,680 in2

The giant hockey puck has a volume of 452,160 cubic inches and a

surface area of 37,680 square inches.

d. ERASERS The dimensions of a rectangular eraser are 2.4 inches

by 4.6 inches by 1 inch. Find the surface area and volume of a

similar eraser that is 5 times as large.

Real-World LinkThe hockey puck that appears to be crashing into the side of the wall at Nationwide Arenain Columbus, Ohio, is about 40 timesthe actual size of a standard puck.

RRT

Example 1 1. The surface area of a rectangular prism is 35 square inches. What is the

surface area of a similar solid that has been enlarged by a scale factor

of 7?

Example 2 2. MODELS The surface area of a ship’s hull is about 11,000 square meters.

What is the surface area, to the nearest tenth, of the hull of a model ship

that is smaller by a scale factor of 1 _

100 ?

Example 3 3. The volume of a cylinder is about 425 cubic centimeters. What is the

volume, to the nearest tenth, of a similar solid that is smaller by a scale

factor of 1 _ 3 ?

Example 4 4. ART STUDIO A sink with a sliding lid in Josh’s art studio measures 16 inches

by 15 inches by 6 inches. A second sink used just for paintbrushes has a

similar shape and is smaller by a scale factor of 1 _ 2 . Find the surface area

and volume of the second sink.

= Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP2.

Example 1 5. The surface area of a rectangular prism is 1,300 square inches. Find the

surface area of a similar solid that is larger by a scale factor of 3.

6. The surface area of a triangular prism is 10.4 square meters. What is the

surface area of a similar solid that is smaller by a scale factor of 1 _ 4 ?

Example 2 77 FOOD A cereal box has a surface area of 280 square inches. What is the

surface area of a similar box that is larger by a scale factor of 1.4?

8. DISPLAYS A glass display box has a surface area of 378 square inches.

How many square inches of glass are used to create a glass display box

with dimensions one-half the original?

Example 3 9. A cone has a volume of 9,728 cubic millimeters. What is the volume of a

similar cone one-eighth the size of the original?

10. A triangular prism has a volume of 350 cubic meters. If the dimensions

are tripled, what is the volume of the new prism?

Example 4 11. ARCHITECTURE The model of a new

18 in.

10 in.

28 in.

apartment building is shown. The

architect plans for the building to

be 144 times the size of the model.

What will be the surface area and

volume of the new building when

it is completed?

Lesson 2C Similar Solids 657

658 Measurement and Proportional Reasoning

12. PUZZLES The world’s largest cube puzzle is in Knoxville, Tennessee. It

measures 6 feet on each side. The scale factor between a standard cube

puzzle and largest puzzle is 1 _ 24

. Find the surface area and volume of the

standard cube puzzle.

1313 Two spheres are similar in shape. The scale factor between the smaller

B sphere and the larger sphere is 3 _ 4 . If the volume of the smaller sphere is

126.9 cubic meters, what is the volume of the larger sphere?

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

14. Two prisms with equal bases are similar.

15. Similar solids have equal volumes.

16. Two cubes are similar.

17. A prism and pyramid are similar.

18. Find the missing

10.5 ft 3 ft

7 ft1.5 ft

y ft

x ft

measure for the

pair of similar solids.

19. Two similar cylinders 18 cm

6 cm

Cylinder A Cylinder B

V = 1,130.4 cm3S.A. = 5,425.92 cm2

are shown.

a. What is the ratio of

their radii?

b. What is the ratio of their

surface areas? volumes?

c. Find the surface area

of Cylinder B.

d. Find the volume of Cylinder A.

Real-World LinkCube Puzzles The world’s largest automated cube puzzle solves itself every 30 seconds. The record time a standard cube puzzle was solved in 2008 was 7.08 seconds.

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20. MEASUREMENT Find the missing values in the table. Use 3.14 for π.

Volume (cm3)

Surface Area (cm2)

Radius (cm)

Cylinder A 3.14 12.57 1

Cylinder B 84.82

Cylinder C 452.52

Test PracticeT

C 21. CHALLENGE A frustum is the solid left

frustum1.5 in.

3 in.

3 in.

6 in.after a cone is cut by a plane parallel

to its base and the top cone is removed.

a. Is the smaller cone that is removed

similar to the original cone? Justify

your response.

b. What is the volume of the smaller

cone? the larger cone? Use 3.14 for π.

c. What is the ratio of the volume of the smaller cone to the volume of

the larger cone?

d. What is the volume of the frustum?

22. E WRITE MATH Explain what happens to the volume of a prism

when its dimensions are tripled.

25. ART Julianna is making a clay figurine of a dog. The dog is 75 centimeters

tall. If she uses a scale of 1 centimeter = 10 centimeters, how tall will the

clay figurine be? (Lesson 2A)

26. SPORTS The table shows the dimensions of the fields used Sport Length

(yards)Width(yards)

Field hockey 60 100

Football 53 1 _ 3 120

Lacrosse 60 110

Soccer 70 115

in various sports. (Lesson 1F)

a. What is the area of the field hockey field in square feet?

b. What is the difference between the area of the soccer

field and the area of the lacrosse field in square feet?

c. If an acre is 43,560 square feet, about how many acres

are all four fields combined?

23. For the similar pyramids, find the ratio

of the surface area of the larger pyramid

to the smaller pyramid.

25 cm 15 cm

A. 5 _ 3 C. 25

_ 9

B. 25 _

15 D. 10

_ 6

24. Two similar prisms have volumes of

4 cubic meters and 864 cubic meters,

respectively. How many times larger

is the second prism?

F. 6 times

G. 16 times

H. 96 times

I. 216 times

Lesson 2C Similar Solids 659