10-810-8Areas and Volumes of Areas and Volumes of

Similar SolidsSimilar Solids

Theorem 10-12Theorem 10-12Similar SolidsSimilar Solids

Similarity RatioSimilarity Ratio

VocabularyVocabulary

Similar SolidsSimilar Solids

Similarity RatioSimilarity Ratio

If similar solids have If similar solids have the same shape, all the same shape, all of their of their corresponding corresponding shapes are shapes are proportional.proportional.

The ratio of The ratio of corresponding linear corresponding linear dimensions of two dimensions of two similar solids.similar solids.

#1 Identifying Similar Solids

Are the two cylinders similar? If so, give the similarity ratio.

1210

6

5

6

5

12

10

Large

Small

6 : 5

65

65

Theorem 10-12

Areas and Volumes of Similar Solids If the similarity ratio of two similar solids is a:b,

then

1. the ratio of their corresponding areas is a2:b2,

and

2. the ratio of their volumes is a3:b3

#2 Finding the Similarity Ratio

Find the similarity ratio of two similar prisms with surface areas 144 m2 and 324 m2.

324

1442

2

b

a

b

a

b

a

2

2

3 : 2324

144

b

a

18

12

b

a

#3 Using a Similarity Ratio The volume of two similar solids are 128 m3 and 250

m3. The surface area of the larger solid is 250 m2. What is the surface area of the smaller solid?

250

128

2

1 V

V

3

3

125

64

b

a

2

2

2

1

5

4

S

S

25

16

2501 S

400025 1 S

21 m 160S

2

2

125

64

5

4

25

16

#4 Using a Similarity Ratio The volume of two similar solids are 5 m3 and 40 m3.

The surface area of the smaller solid is 4 m2. What is the surface area of the larger solid?

40

5

2

1 V

V

3

3

40

5

b

a

23

23

2

1

)40(

)5(

S

S

70.11

92.24

2

S

80.4692.2 2 S

22 m 16S

70.11

92.2

#5 Using a Similarity Ratio Two similar square pyramids have volumes of 48 cm3

and 162 cm3. The surface area of the larger pyramid is 135 cm2. Find the surface area of the smaller pyramid.

162

48

2

1 V

V

3

3

27

8

b

a

2

2

2

1

3

2

S

S

9

4

1351 S

5409 1 S2

1 cm 60S

6

6

27

8

3

2

9

4

#6 Using a Similarity Ratio Two similar square pyramids have surface areas of 52

cm2 and 208 cm2. The volume of the larger pyramid is 192 cm3. Find the volume of the smaller pyramid.

208

52

2

1 S

S

16

4

b

a

3

3

2

1

2

1

V

V

8

1

1921 V

1928 1 V

31 cm 24V

13

13

16

4

2

1

4

2

8

1

Assignment

Page 568-569 #1-16