Course 2

10-5 Changing Dimensions

Warm UpFind the surface area of each figure to the nearest tenth. Use 3.14 for .

1. a cylinder with radius 5 ft and height 11 ft

2. a sphere with diameter 6 ft

3. a rectangular prism 9 ft by 14 ft by 6 ft

502.4 ft2

113.0 ft2

528 ft2

Course 2

10-5 Changing Dimensions

EQ: How do I find the volume and surface area of similar three-dimensional figures?

M7G3.a Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts; M7G3.b Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures;

Course 2

10-5 Changing Dimensions

Recall that similar figures are proportional. The surface areas of similar three-dimensional figures are also proportional. To see this relationship, you can compare the areas of corresponding faces of similar rectangular prisms.

Course 2

10-5 Changing Dimensions

Area of front ofsmaller prism

Area of front of larger prism

3 · 5 6 · 10

15 (3 · 2) · (5 · 2)

(3 · 5) · (2 · 2)

15 · 22

Each dimension has a scale factor of 2.

A scale factor is a number that every dimension of a figure is multiplied by to make a similar figure.

Remember!

Course 2

10-5 Changing Dimensions

The area of the front face of the larger prism is 22 times the area of the front face of the smaller prism. This is true for all of the corresponding faces. Thus it is also true for the entire surface area of the prisms.

SURFACE AREA OF SIMILAR FIGURES

The surface area of a three-dimensional figure A is equal to the surface area of a similar figure B times the square of the scale factor of figure A.

surface area of figure A

surface area of figure B

(scale factor)2= •

Course 2

10-5 Changing Dimensions

The surface area of a box is 35 in2. What is the surface area of a larger, similar box that is larger by a scale factor of 7?

Additional Example 1A: Finding the Surface Area of a Similar Figure

S = 35 · 72 Multiply by the square of the scale factor.

S = 35 · 49 Evaluate the power.

S = 1,715 Multiply.

The surface area of the larger box is 1,715 in2.

Course 2

10-5 Changing Dimensions

The surface area of a box is 1,300 in2. Find the surface area of a smaller, similar box that is

smaller by a scale factor of .

Additional Example 1B: Finding the Surface Area of a Similar Figure

12

S = 1,300 · 12

2

S = 1,300 · 14

S = 325

The surface area of the smaller box is 325 in2.

Multiply by the square of the scale factor.

Evaluate the power.

Multiply.

Course 2

10-5 Changing Dimensions

Check It Out: Example 1A

S = 50 · 32 Multiply by the square of the scale factor.

S = 50 · 9 Evaluate the power.

S = 450 Multiply.

The surface area of the larger box is 450 in2.

The surface area of a box is 50 in2. What is the surface area of a larger, similar box that is larger by a scale factor of 3?

Course 2

10-5 Changing Dimensions

13

S = 1,800 · 13

2

S = 1,800 · 19

S = 200

The surface area of the smaller box is 200 in2.

Multiply by the square of the scale factor.

Evaluate the power.

Multiply.

Check It Out: Example 1B

The surface area of a box is 1,800 in2. Find the surface area of a smaller, similarly shaped box

that has a scale factor of .

Course 2

10-5 Changing Dimensions

The volumes of similar three-dimensional figures are also related.

3 ft

2 ft

1 ft2 ft

6 ft

4 ft

Volume of smaller box

Volume oflarger box

2 · 3 · 1

64 · 6 · 2

(2 · 2) · (3 · 2) · (1 · 2)

(2 · 3 · 1) · (2 · 2 · 2)

6 · 23

Each dimensionhas a scale factor of 2.

The volume of the larger box is 23 times the volume of the smaller box.

Course 2

10-5 Changing Dimensions

VOLUME OF SIMILAR FIGURES

The volume of three-dimensional figure A is equal to the volume of a similar figure B times the cube of the scale factor of figure A.

volume of figure A

•volume of figure B

= (scale factor)3

Course 2

10-5 Changing Dimensions

The volume of a child’s swimming pool is 28 ft3. What is the volume of a similar pool prism that is larger by a scale factor of 4?

Additional Example 2: Finding Volume Using Similar Figures

V = 28 · 43 Multiply by the cube of the scale factor.

V = 28 · 64 Evaluate the power.

V = 1,792 ft3 Multiply.

Estimate V ≈ 30 · 60 Round the measurements.

= 1,800 The answer is reasonable.

Course 2

10-5 Changing Dimensions

Check It Out: Example 2

The volume of a small hot tube is 48 ft3. What is the volume of a similar hot tub that is larger by a scale factor of 2?

V = 48 · 23 Use the volume of the smaller prismand the cube of the scale factor.

V = 48 · 8 Evaluate the power.

V = 384 ft3 Multiply.

Estimate V ≈ 50 · 8 Round the measurements.

= 400 The answer is reasonable.

Course 2

10-5 Changing Dimensions

The sink in Kevin’s workshop measures 16 in. by 15 in. by 6 in. Another sink with a similar shape is larger by a scale factor of 2. There are 231 in3 in 1 gallon. Estimate how many more gallons the larger sink holds.

Additional Example 3: Problem Solving Application

Course 2

10-5 Changing Dimensions

Additional Example 3 Continued

11 Understand the Problem

Rewrite the question as a statement.

• Compare the capacities of two similar sinks, and estimate how much more water the larger sink holds.

List the important information:• The smaller sink is 16 in. x 15 in. x 6 in.

• The larger sink is similar to the small sink by a scale factor of 2.

• 231 in3 = 1 gal

Course 2

10-5 Changing Dimensions

Additional Example 3 Continued

22 Make a Plan

You can write an equation that relates the volume of the large sink to the volume of the small sink. The convert cubic inches to gallons to compare the capacities of the sinks.

Volume of large sink = Volume of small sink · (a scale factor)3

Course 2

10-5 Changing Dimensions

Additional Example 3 Continued

Solve33Volume of small sink = 16 x 15 x 6 = 1,440 in3

Convert each volume into gallons:

Volume of large sink = 1,440 x 23 = 11,520 in3

1,440 in3 x ≈ 6 gallons1 gal 231 in3

11,520 in3 x ≈ 50 gallons1 gal 231 in3

Subtract the capacities: 50 gal – 6 gal = 44 gal

The large sink holds about 44 gallons more than the small sink.

Course 2

10-5 Changing Dimensions

Look Back44

Double the dimensions of the small sink and find the volume:32 x 30 x 12 = 11,520 in3. Subtract the volumes of the two sinks:11,520 – 1,440 = 10,080 in3. Convert this measurement to gallons:

10,080 x ≈ 44 gal.

Additional Example 3 Continued

1 gal 231 in3

Course 2

10-5 Changing Dimensions

The bath tub in Ravina’s house measures 46 in. by 36 in. by 24 in. Another bath tub with a similar shape is

smaller by a scale factor of . There are

231 in3 in 1 gallon. Estimate how many more gallons the larger bath tub holds.

Check It Out: Example 3

1 2

Course 2

10-5 Changing Dimensions

Check It Out: Example 3 Continued

11 Understand the Problem

Rewrite the question as a statement.

• Compare the capacities of two similar tubs, and estimate how much more water the larger tub holds.

List the important information:• The larger tub is 46 in. x 36 in. x 24 in.

• 231 in3 = 1 gal

• The smaller tub is similar to the larger tub by a scale factor of . 1

2

Course 2

10-5 Changing Dimensions

Check It Out: Example 3 Continued

22 Make a Plan

You can write an equation that relates the volume of the small tub to the volume of the large tub. The convert cubic inches to gallons to compare the capacities of the tubs.

Volume of small tub = Volume of large tub · (a scale factor)3

Course 2

10-5 Changing Dimensions

Check It Out: Example 3 Continued

Solve33Volume of large tub = 46 x 36 x 24 = 39,744 in3

Convert each volume into gallons:

Volume of small tub = 39,744 x 0.53 = 4,968 in3

39,744 in3 x ≈ 172 gallons1 gal 231 in3

4,968 in3 x ≈ 22 gallons1 gal 231 in3

Subtract the capacities: 172 gal – 22 gal = 150 gal

The large tub holds about 150 gallons more than the small tub.

Course 2

10-5 Changing Dimensions

Look Back44

Half the dimensions of the large tub and find the volume:23 x 18 x 12 = 4,968 in3. Subtract the volumes of the two tubs:39,744 – 4,968 = 34,776 in3. Convert this measurement to gallons:

34,776 x ≈ 150 gal.

Check It Out: Example 3 Continued

1 gal 231 in3

Course 2

10-5 Changing Dimensions

Lesson Quiz: Part I

Given the scale factor, find the surface area to the nearest tenths of the similar prism.

1. The scale factor of the larger of two similar

triangular prisms is 8. The surface area of the

smaller prism is 18 ft2.

2. The scale factor of the smaller of two similar

triangular prisms is . The surface area of the

larger prism is 600 ft2.

66.7 ft2

1,152 ft2

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Course 2

10-5 Changing Dimensions

Lesson Quiz: Part II

Given the scale factor, find the volume of the similar prism.

3. The scale factor of the larger of two similar

rectangular prisms is 3. The volume of the

smaller prism is 12 cm3.

4. A food storage container measures 6 in. by 10

in. by 2 in. A similar container is reduced by a

scale factor of . Estimate how many more

gallons the larger container holds.

324 cm3

About 0.5 gal

12