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Pearson Geometry Common Core, 2011

PG

Pearson Geometry Common Core, 2011 View details

7. Areas and Volumes of Similar Solids

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Exercise 37 Page 748

Given the ratio of volumes for two similar solids, we can use the cube root to get the scale factor.

Similarity Ratio	Ratio of Surface Areas	Ratio of Volumes
5:8	25:64	125:512

Practice makes perfect

The ratio of two similar solid's volumes is 125 and 512. We want to find the similarity ratio of the two solids and the ratio of their surface areas.

Similarity Ratio

Let's use the Areas and Volumes of Similar Solids Theorem to recall the relationship between the scale factor and volume.

Areas and Volumes of Similar Solids Theorem
If the scale factor of two similar solids is ab or a:b then the ratio of their volumes is a^3b^3 or a^3:b^3.

To find the similarity ratio we will cube root the ratio of their volumes.

sqrt(a^3/b^3)

a^3= 125 and b^3= 512

sqrt(125/512)

sqrt(a/b)=sqrt(a)/sqrt(b)

sqrt(125)/sqrt(512)

Calculate root

5/8

\dfrac a b = a:b

5 : 8

The similarity ratio of these two solids is 58 and can also be written as 5:8.

Ratio of Surface Areas

Let's use the Areas and Volumes of Similar Solids Theorem again to recall the relationship between the scale factor and area.

Areas and Volumes of Similar Solids Theorem
If the scale factor of two similar solids is ab or a:b then the ratio of their areas is a^2b^2 or a^2:b^2.

To find the ratio of their surface areas we will substitute in the scale factor values.

a^2/b^2

a= 5 and b= 8

5^2/8^2

Calculate power

25/64

\dfrac a b = a:b

25 : 64

The ratio of their areas is 2564 and can also be written as 25:64.

Similarity Ratio	Ratio of Surface Areas	Ratio of Volumes
5:8	25:64	125:512

Subchapter links

Lesson Check p.745

Practice and Problem-Solving Exercises p.746-749

Standardized Test Prep p.749

Mixed Review p.749