9-3B Volume of Similar Solids

What is the relationship between the similarity ratio of two similar solids and the

volumes of those 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

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

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

Suppose that two similar animals have a similarity ratio of 3/2. The smaller one weighs 10 lb. What does the larger one weigh?

Volume

8

27

2

33

lb75.33108

27

Similarity Ratio

(any part)

1

1

2

SurfaceArea Ratio

Volume Ratio

1

4

1

22

1

8

1

23

Theorem

The ratio of the volumes of two similar solids is the cube of their similarity ratio.

• What is the relationship between the similarity ratio of two similar solids and the volumes of those solids?

The similarity ratio is cubed to find the similarity ratio of the volumes.