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GEOMETRIC SOLIDS

Similar Solids

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SIMILAR SOLIDSDefinition: 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

824

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

9

Are these solids similar?

Solution:

All of the corresponding lengths have the same scale factor, therefore the figures are similar.

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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

104

8

Surface Area = LA + B 108 + 40 = 148

52

43.5

Ratio of sides = 2: 1

Ratio of LA: 108:27 = 4:1Ratio of Base Area = 40:10 = 4:1Ratio of Surface Area = 148:37 = 4:1

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Surface Area = LA + B 27+ 10 = 37

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 = πr2 h = π(92) (15) = 1215π

V= πr2 h = π(62)(10) = 360π

Ratio of volumes= 1215 π : 360 π = 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. If the original volume was 2000 cubic feet, 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.