lesson 9-5: similar solids 1 lesson 9-5 similar solids
TRANSCRIPT
Lesson 9-5: Similar Solids 1
Lesson 9-5
Similar Solids
Lesson 9-5: Similar Solids 2
Similar SolidsTwo 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
Similar Solids
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.
123
6
82
4
Length: 12 = 3 width: 3 height: 6 = 3 8 2 2 4 2
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
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
Are these solids similar?
Example:
Solution:
Lesson 9-5: Similar Solids 6
8
18
4
6
Corresponding ratios are not equal, so the figures are not similar.
Are these solids similar?
Solution:
8 2:
4 118 3
:6 1
radius
height
Example:
Lesson 9-5: Similar Solids 7
What is the scale factor for the two rectangles?The ratio of the areas can be written as
What happens to the area when the lengths of the sides of a rectangle are doubled?
Scale Factor and Area
Ratio of sides = 1: 2
Ratio of areas = 1: 4
1: 212: 22
Ratio of surface areas: 297:132 = 9:4 = 32: 22
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.
8
9
9
12
Similar Solids and Ratios of Areas
Surface Area = B + L.A.= 9(9) + (9 + 9 + 9 + 9)(12)/2 = 81 +216 = 297
66
8
Surface Area = B + L.A.= 6(6) + (6 + 6 + 6 + 6)(8)/2= 36 + 96 = 132
Ratio of sides = 3: 2
Lesson 9-5: Similar Solids 9
The scale factor for the two prisms isThe ratio of the surface areas can be written asThe ratio of the volumes can be written as
What happens to the surface area and volume when the lengths of the sides of a prism are doubled?
Scale Factor and Volume
Ratio of sides = 1: 2
Ratio of areas = 1: 4
Ratio of volumes = 1: 81: 212: 22
13: 23
Lesson 9-5: Similar Solids 10
9
15
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.
6
10Ratio of heights = 3:2
V = r2h = (92) (15) = 1215
V= r2h = (62)(10) = 360
Ratio of volumes: 1215: 360 = 27:8 = 33: 23
These two solids are similar.
a.The scale factor is
b.The ratio of areas is
c.The ratio of volumes is
Example 1:
Lesson 9-5: Similar Solids 11
18 m6 m
18: 6 = 3: 1
182: 62 = 32: 12 = 9:1
183: 63 = 33: 13 = 27:1
These two solids are similar.
If the radius of the larger cone is 6 m, what is the radius of the smaller cone?
Example 2:
Lesson 9-5: Similar Solids 12
18 m6 m
Solution: Write a proportion.
18 6
618 6 6
2
rr
r m
These two solids are similar.
If the lateral area of the smaller cone is 12, what is the lateral area of the larger cone?
Example 3:
Lesson 9-5: Similar Solids 13
18 m6 m
Solution: Write a proportion. Use ratio of AREAS.
2
9
1 121 108
108
LA
LA
LA m
These two solids are similar.
If the volume of the larger cone is 96 , what is the volume of the smaller cone?
Example 4:
Lesson 9-5: Similar Solids 14
18 m6 m
Solution: Write a proportion. Use ratio of VOLUMES.
3
27 96
127 1 96
32
9
VV
V m
Volume of larger is 27 times volume of smaller!