12-8 Congruent and Similar Solids

You compared surface areas and volumes of spheres.

• Identify congruent or similar solids.

• Use properties of similar solids.

Similar Cylinders• Are the cylinders similar?• What is their similarity

ratio?• Find the volume of each

cylinder.• What is the ratio of their

volumes?1

3

3

9

V = π(1)23

V = 3π

V = π(3)2(9)

V = 81π

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Similar solids have exactly the same shape, but not necessarily the same size.All spheres are similar and all cube are similar.In similar solids, the corresponding linear measures (like height or radius) have equal ratios.The common ratio is called the scale factor.

p. 896

Congruent solids have exactly the same shape and the same size.Congruent solids are similar solids that have a scale factor of 1:1

Similar Animals• The weight of an object of a given density is

proportional to its volume. Suppose the measurements of one animal are four times the measurements of another, and the two animals are geometrically similar. How many times greater is the weight of the larger animal than the weight of the smaller?

• Similarity ratio 4/1 • V=(4)3=64

Two years ago, Josh bought a puppy that weighed 2 lb. Since then, it has tripled in length. About how much does the dog weigh now?

The similarity ratio is 3/1. Ratio of the volumes would be (3/1)3= 27Weight is proportional to volume, so the dog

now weighs about 2 ×27 = 54 lbs

What is the ratio of the volumes of the two cubes shown?

8

1

20 cm

40 cm

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

Compare the ratios between the corresponding parts of the cones.

Answer: Since the ratios are not the same, the cones are neither similar nor congruent.

Simplify.

Substitution

Substitution

=1

A. similar

B. congruent

C. neither

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

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Two similar cones have radii of 9 inches and 12 inches. What is the ratio of the volume of the smaller cone to the volume of the larger cone?

First, find the scale factor.

Write a ratio comparing the radii.

If the scale factor is , then the ratio of the volumes is .

Answer: So, the ratio of the volume is 27:64.

A. 1:3

B. 1:9

C. 1:27

D. 1:81

Two similar cones have radii of 5 inches and 15 inches. What is the ratio of the volume of the smaller cone to the volume of the larger cone?

SOFTBALLS The softballs shown are similar spheres. Find the radius of the smaller softball if the radius of the larger one is about 1.9 cubic inches.

Understand You know the volume of the softballs.Plan Use Theorem 12.1 to write a ratio

comparing the volumes. Then find the scale factor and use it to find r.

Solve Write a ratio comparing volumes.

Simplify.

Write as .

=

≈

Ratio of radii Scale factor

Find the cross products.

Solve for r.

Answer: So, the radius of the smaller softball is about 1.45 inches.

r ≈ 1.45

The ratio of the volumes of two similar cylinders is 125/27. What is the ratio of their radii?

3

?

?

27

125

??

Sim ratio

Vol ratio

3

5

A. 2 in.

B. 3 in.

C. 4 in.

D. 5 in.

CONTAINERS The containers below are similar cylinders. Find the height h of the smaller container.

Similarity Ratio

(any part)

1

1

2

SurfaceArea Ratio

Volume Ratio

1

4

1

22

1

8

1

23