Transcript 10-8 Areas and Volumes of Similar Solids

10-8
Areas and Volumes of
Similar Solids
Theorem
10-12
Similar Solids
Similarity Ratio
Vocabulary

Similar Solids

Similarity Ratio
If similar solids have
the same shape, all of
their corresponding
shapes are
proportional.
 The ratio of
corresponding linear
dimensions of two
similar solids.

#1 Identifying Similar Solids
Are the two cylinders similar? If so,
give the similarity ratio.
Small
6
Large

5
12
10
10 5

12 6
5 5

6 6
5:6
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.
a2 a

2
b
b
a 2 144

2
b
324
a
144

b
324
a 12

b 18
2:3
#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?
V1 128  2 64


V2 250  2 125
S1 4
16
 2 
25
S2 5
4
a 3 64

3
5
b
125
S1
16

250 25
2
25S1  4000
S1  160m2
#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?
V1
5

V2 40
a 35
3
b
40
S1
(3 5 ) 2 2.92

 3
S2 ( 40) 2 11.70
4
2.92

S 2 11.70
2.92S2  46.80
S2  16 m2
#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.
V1 48  6
8


V2 162  6 27
S1 2
4
 2 
9
S2 3
2
a 38

3
b
27 3
S1
4

135 9
2
9S1  540
S1  60 cm2
#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.
S1
52  13 4


S 2 208 13 16
2 1
a
4
 

b
16 4 2
3
V1 1
1
 3 
8
V2 2
V1
1

192 8
8V1  192
V1  24 cm3
Assignment
Page 568-569
#1-16