12-7 Similar Solids

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Warm UpClassify each polygon.

1. A square has a side 4 cm, a similar larger square has a length of 20 cm. Provide the perimeter ration.

2. Use problem 1 to provide the length ratio.

3. Now the answer in problem 2 to provide the scale factor (unit ratio):

Perimeter Ratio 16:80

Length Ratio: 4:20

Unit Ration: 1:5

12.7 Similar Solids

1:5

Find and Use the scale factor of Similar Solids

Use Similar Solids to solve real-life problems.

Objectives

12.7 Similar Solids

Similar Solids

Vocabulary

12.7 Similar Solids

12.7 Similar Solids

Similar Solids

Two solids of the same type with equal ratios of corresponding linear measures (such as heights or radii) are called similar solids.

12.7 Similar Solids

Similar solids NOT similar solids

12.7 Similar Solids

Similar Solids & Corresponding Linear Measures

123

6

82

4

Length: 12 = 3 width: 3 height:6 = 3

Notice that all ratios for corresponding measures are equal in similar solids. The reduced ratio is called the

“scale factor”.

To compare the ratios of corresponding side or other linear lengths, write the ratios as fractions in simplest terms.

8 2 2 4 2

Example 1A: Are these Solids Similar

12.7 Similar Solids

16

12

8

612

9

Example 1A: Are these Solids Similar

12.7 Similar Solids

16

12

8

612

9

All corresponding ratios are equal, so the figures are similar

16 4:12 38 4:6 312 4:9 3

length

width

height

Solution:

Example 1B: Decide if the solids are Similar.

12.7 Similar Solids

8

18

4

6

Example 1B: Classifying Three-Dimensional Figures

12.7 Similar Solids

8

18

4

6

Corresponding ratios are not equal, so the figures are not similar

Solution:

8 2:4 1

radius

18 3:6 1

height

12.7 Similar Solids

• If two similar solids have a scale factor of a : b,

then corresponding areas have a ratio of a2: b2.

• This applies to lateral area, surface area, or base area.

• Length/Perimeter ratio a:b

• Area Ratios a2 : b 2

Similar Solids and Ratio of Areas

12.7 Similar Solids

10

4

8

Surface Area = base +lateral = 40 + 108

= 148

52

4

3.5

Surface Area =base +lateral = 10 + 27

= 37

Ratio of sides = 2:1

7

Ratio of surface areas:= 148:37 = 4:1 = 22: 12

Example 1C: Similarity Ratios

Similar Solids and Volume Ratios

12.7 Similar Solids

• If two similar solids have a scale factor of a : b, then their volumes have a ratio of a3 : b3.

Length/Perimeter Ratios a:b

Area Ratios a2: b2

Volume Ratios a3: b3

Example 1D: Similar Solids and Volume Ratios

12.7 Similar Solids

9

15

6

10

Ratio of heights = 3:2

V = r2h = (92) (15) = 1215

V= r2h = (62)(10) = 360

Ratio of volumes: = 1215:360 = 27:8 = 33: 23

1. The following solids are similar. Provide the length, area and volume ratios.

Lesson Quiz: Part I

Length ratios (a:b) = 3:6 = 1:2

Area ratios: (a2:b2) = 1:4

Volume ratios: (a3:b3) = 1:8

12.7 Similar Solids

2. The following solids are similar. Provide the length, area and volume ratios.

Lesson Quiz: Part II

Length ratios (a:b) = 12:4 = 3:1

Area ratios: (a2:b2) = 9:1

Volume ratios: (a3:b3) = 27:1

12.7 Similar Solids

2. The following solids are similar. Provide the ratios of the length and area.

Lesson Quiz: Part III

Length ratios (a:b) = 3:6 = 1:2

Area ratios: (a2:b2) = 1:4

Volume ratios: (a3:b3) = 27:216

12.7 Similar Solids

Take the cube root of the volume to get the

length ratio.

2. The following solids are similar. Provide the ratios of the length and area.

Lesson Quiz: Part III

Length ratios (a:b) = 3:5

Area ratios: (a2:b2) = 9:25

Volume ratios: (a3:b3) = 27:125

12.7 Similar Solids

Take the cube root of the volume to get the

length ratio.

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12.7 Similar Solids