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GEOMETRIC SOLIDS
Similar Solids
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SIMILAR SOLIDS
Definition: 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 otherlinear 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”.
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EXAMPLE:
16 4:12 38 4:6 312 4:9 3
length
width
height
16
12
8
612
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Are these solids similar?
Solution:
All of the corresponding lengths have the same scale factor, therefore the figures are similar.
6
8 2:4 118 3:6 1
radius
height
8
18
4
6
Corresponding ratios are not equal, so the figures are not similar.
Are these solids similar?
Solution:
Example:
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• Base Area
SIMILAR SOLIDS AND RATIOS OF AREAS
8
10
4
8
Surface Area = base + lateral = 40 + 108 = 148
52
4
3.5
Surface Area = base + lateral = 10 + 27 = 37
Ratio of sides = 2: 1
Ratio of surface areas = 148:37 = 4:1 = 22: 12
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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.
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SIMILAR SOLIDS AND RATIOS OF VOLUMES
9
15
6
10
Ratio of heights = 3:2
V = r2 h = (92) (15) = 1215
V= r2 h = (62)(10) = 360
Ratio of volumes= 1215 : 360 = 27 : 8 = 33
: 23
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SIMILAR SQUARES EXAMPLE
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CHANGE IN DIMENSION EXAMPLE
• The dimensions of a water touch tank at the local aquarium are doubled. What is the volume of the new tank?
FINDING THE SCALE FACTOR OF SIMILAR SOLIDS
• To find the scale factor of the two cubes, find the ratio of the two volumes.
Write ratio of volumes.
Use a calculator to take the cube root.Simplify.
So, the two cubes have a scale factor of 2:3.
COMPARING SIMILAR SOLIDS
• Swimming pools. Two swimming pools are similar with a scale factor of 3:4. The amount of chlorine mixture to be added is proportional to the volume of water in the pool. If two cups of chlorine mixture are needed for the smaller pool, how much of the chlorine mixture is needed for the larger pool?
SOLUTION:
• Using the scale factor, the ratio of the volume of the smaller pool to the volume of the larger pool is as follows:
• The ratio of the volumes of the mixture is 1:2.4. The amount of the chlorine mixture for the larger pool can be found by multiplying the amount of the chlorine mixture for the smaller pool by 2.4 as follows: 2(2.4) = 4.8 c.
• So the larger pool needs 4.8 cups of the chlorine mixture.
