Similar Solids

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Lesson 9-5
Similar Solids
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Similar Solids

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

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Similar Solids

• Similar solids

• NOT similar solids

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Similar Solids Corresponding Linear Measures

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

Length 12 3 width 3
height 6 3 8 2
2 4
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Notice that all ratios for corresponding measures
are equal in similar solids. The reduced ratio
is called the scale factor.
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Example
Are these solids similar?
Solution
All corresponding ratios are equal, so the
figures are similar
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Example
Are these solids similar?
Solution
Corresponding ratios are not equal, so the
figures are not similar.
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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
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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.

Ratio of sides 3 2
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Surface Area B L.A. 6(6) (6 6 6
6)(8)/2 36 96 132
Surface Area B L.A. 9(9) (9 9 9
9)(12)/2 81 216 297
Ratio of surface areas 297132 94 32 22
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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
1 2
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
12 22
13 23
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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.

Ratio of heights 32
V ?r2h ? (92) (15) 1215 ?
V ?r2h ? (62)(10) 360 ?
Ratio of volumes 1215? 360? 278 33 23
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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
182 62 32 12 91
183 63 33 13 271
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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.
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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.
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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.