1. Lesson 9-5: Similar Solids

2. Lesson 9-5: Similar Solids
Two solids of the same type with equal ratios of corresponding linear measures (such as heights or radii) are called similar solids.
Lesson 9-5: Similar Solids

3. Lesson 9-5: Similar Solids
NOT similar solids
Lesson 9-5: Similar Solids

4. Similar Solids & Corresponding Linear Measures
To compare the ratios of corresponding side or other linear lengths, write the ratios as fractions in simplest terms. 12 3 6 8 2 4
Length: 12 = width: height: 6 = 3
Notice that all ratios for corresponding measures are equal in similar solids. The reduced ratio is called the “scale factor”.
Lesson 9-5: Similar Solids

5. Lesson 9-5: Similar Solids
Example:
Are these solids similar?
Solution: 16 12 8 6 9
All corresponding ratios are equal, so the figures are similar
Lesson 9-5: Similar Solids

6. Lesson 9-5: Similar Solids
Example:
Are these solids similar? 8 18 4 6
Solution:
Corresponding ratios are not equal, so the figures are not similar.
Lesson 9-5: Similar Solids

7. Lesson 9-5: Similar Solids
Scale Factor and Area
What happens to the area when the lengths of the sides of a rectangle are doubled?
Ratio of sides = 1: 2
Ratio of areas = 1: 4
What is the scale factor for the two rectangles?
The ratio of the areas can be written as
1: 2
12: 22
Lesson 9-5: Similar Solids

8. Similar Solids and Ratios of Areas
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. 6 8
Ratio of sides = 3: 2 12 9 9
Surface Area = B + L.A.
= 6(6) + ( )(8)/2
= = 132
Surface Area = B + L.A.
= 9(9) + ( )(12)/2
= = 297
Ratio of surface areas: 297:132 = 9:4 = 32: 22

9. Lesson 9-5: Similar Solids
Scale Factor and Volume
What happens to the surface area and volume when the lengths of the sides of a prism are doubled?
Ratio of sides = 1: 2
Ratio of areas = 1: 4
Ratio of volumes = 1: 8
The scale factor for the two prisms is
The ratio of the surface areas can be written as
The ratio of the volumes can be written as
1: 2
12: 22
13: 23
Lesson 9-5: Similar Solids

10. Similar Solids and Ratios of Volumes
If two similar solids have a scale factor of a : b, then their volumes have a ratio of a3 : b3. 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
Lesson 9-5: Similar Solids

11. Lesson 9-5: Similar Solids
Example 1:
These two solids are similar.
The scale factor is
The ratio of areas is
The ratio of volumes is
18: 6 = 3: 1
18 m
6 m
182: 62 = 32: 12 = 9:1
183: 63 = 33: 13 = 27:1
Lesson 9-5: Similar Solids

12. Lesson 9-5: Similar Solids
Example 2:
These two solids are similar.
If the radius of the larger cone is 6 m, what is the radius of the smaller cone?
Solution: Write a proportion.
18 m
6 m
Lesson 9-5: Similar Solids

13. Lesson 9-5: Similar Solids
Example 3:
These two solids are similar.
If the lateral area of the smaller cone is 12, what is the lateral area of the larger cone?
Solution: Write a proportion. Use ratio of AREAS.
18 m
6 m
Lesson 9-5: Similar Solids

14. Lesson 9-5: Similar Solids
Volume of larger is 27 times volume of smaller!
Example 4:
These two solids are similar.
If the volume of the larger cone is 96 , what is the volume of the smaller cone?
Solution: Write a proportion. Use ratio of VOLUMES.
18 m
6 m
Lesson 9-5: Similar Solids