Volume Ratios of Similar Solids

Theorem 87-1 gives the relationship between similar polygons with sides that are in an 𝑎: 𝑏 ratio. Theorem 99-1 provides a similar rule for three-dimensional solids.

Geometry Lesson 99

Theorem 99-1 - If two similar solids have a scale factor of 𝒂: 𝒃 then, 1. the ratio of the perimeters of their corresponding faces is a:b. 2. the ratio of the areas of their corresponding faces is a 𝒂𝟐: 𝒃𝟐.

3. the ratio of their volumes is 𝒂𝟑: 𝒃𝟑.

Geometry Lesson 99

Notice that the first two parts of this theorem are nearly identical to the two parts of Theorem 87-1.

Geometry Lesson 99

Two similar rectangular pyramids have a scale factor of 3:2. Determine the perimeter of the smaller pyramid’s base. SOLUTION First, determine the perimeter of the larger pyramid’s base.

𝑃 = 7.5 + 7.5 + 3 + 3 𝑃 = 21

By Theorem 99-1, the perimeter of the smaller pyramid’s base is in a 3:2 ratio with the perimeter of the larger pyramid’s base.

21

𝑃=3

2

𝑃 = 14 So the perimeter of the smaller pyramid’s base is 14 inches.

Geometry Lesson 99

The second part of Theorem 99-1 implies that the total surface area of two similar solids with a scale factor of 𝑎: 𝑏 will be 𝑎2: 𝑏2.

Geometry Lesson 99

The surface area of the smaller pyramid shown is 54 square centimeters. What is the surface area of the larger pyramid? SOLUTION Since the pyramids are similar, we can apply Theorem 99-1. From the dimensions given, the ratio of the solids’ sides is 15:6, or 5:2. Apply Theorem 99-1.

52

22=

𝐴

54

𝐴 = 337.5 The surface area of the larger pyramid is 337.5 square centimeters.

Geometry Lesson 99

a. The two cylinders shown are similar. If the volume of the smaller cylinder is 38 cubic feet, what is the volume of the larger cylinder? SOLUTION Looking at the radii, we can see that the scale factor is 3:2.

By Theorem 99-1, the ratio of their volumes will be 33

23.

Write a proportion.

33

23=

𝑉

38

𝑉 = 128.25 The volume of the larger cylinder is 128.25 cubic feet.

Geometry Lesson 99

b. Prove the third part of Theorem 99-1 for any pair of similar pyramids. SOLUTION Suppose similar pyramids K and L have base areas 𝐵𝐾 and 𝐵𝐿 and height ℎ𝐾 and ℎ𝐿 . Since the base areas and heights are of corresponding parts, they are in the ratios 𝑎2: 𝑏2 and 𝑎: 𝑏, respectively. Apply the formula for volume of a pyramid.

𝑉𝐾 =1

3𝐵𝐾ℎ𝐾 Volume of a pyramid

𝑉𝐾 =1

3

𝑎2

𝑏2𝐵𝐿

𝑎

𝑏ℎ𝐿 Substitute.

𝑉𝐾 =𝑎2

𝑏2𝑎

𝑏

1

3𝐵𝐿ℎ𝐿 Simplify.

𝑉𝐾 =𝑎3

𝑏3𝑉𝐿 Substitute.

Geometry Lesson 99

A proposed crew capsule for space exploration is shaped like a square pyramid. A scale model of the capsule has base sides of 26 inches each and a height of 15 inches. Determine the volume of the actual capsule, which will be dilated by a factor of 9, to the nearest cubic foot. SOLUTION The scale factor is 9:1. First, find the volume of the square pyramid. The area of the base is 676, and the height is 15.

𝑉 =1

3𝐵ℎ

𝑉 =1

3676 15

𝑉 = 3380

Now apply Theorem 99-1 by writing a proportion.

93

13=

𝑉

3380

𝑉 = 2464020

This gives the volume in cubic inches.

There are 12 inches in a foot, so to find the volume in cubic feet, divide by 123. The actual capsule will have a volume of approximately 1426 cubic feet.

Geometry Lesson 99

These two similar right triangular prisms have a scale factor of 3:4. a. Determine the perimeter of the larger prism’s base. b. If the surface area of the smaller prism is 182 square inches, find the surface area of the larger prism. c. If the volume of the smaller prism is 323 cubic inches, find the volume of the larger prism.

Geometry Lesson 99

These two similar right triangular prisms have a scale factor of 3:4.

a. Determine the perimeter of the larger prism’s base.

b. If the surface area of the smaller prism is 182 square inches, find the surface area of the larger prism.

c. If the volume of the smaller prism is 323 cubic inches, find the volume of the larger prism.

d. Space Exploration: The first booster stage of the Saturn 1B moon rocket consists of a cylindrical tank section and five rocket motors. In a 1:50 scale model of the booster stage, the tank is 59 centimeters high with a diameter of 20 centimeters. Determine the volume of the actual booster stage, to the nearest cubic meter.

Geometry Lesson 99

d. Space Exploration: The first booster stage of the Saturn 1B moon rocket consists of a cylindrical tank section and five rocket motors. In a 1:50 scale model of the booster stage, the tank is 59 centimeters high with a diameter of 20 centimeters. Determine the volume of the actual booster stage, to the nearest cubic meter.

Geometry Lesson 99

Page 644

Lesson Practice (Ask Mr. Heintz)

Page 645

Practice 1-30 (Do the starred ones first)

Geometry Lesson 99