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# 7.7: Change of Dimensions

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Determine the Surface Area and Volume of similar polyhedra.

## Similar or Not Similar

Two solids of the same type with equal ratios of corresponding linear measures are called similar solids. Corresponding linear measures means matching measurements, such as heights or widths or radii. Equal ratios of these measurements means that the fraction relating both heights is the same as the fraction relating both widths.

To be similar, figures need to have corresponding linear measures that are in proportion to one another. If these linear measures are not in proportion, the figures are not similar.

Example 1

Are these two figures similar?

If the figures are similar, all ratios for corresponding measures must be the same.

The ratios are:

width(w)height(h)depth(d)=69=23=1421=23=812=23\begin{align*}\text{width} (w) &= \frac{6}{9} = \frac{2}{3}\\ \text{height} (h) &= \frac{14}{21} = \frac{2}{3}\\ \text{depth} (d) &= \frac{8}{12} = \frac{2}{3}\end{align*}

Since the 3 ratios are equal, you can conclude that the figures are similar.

A ratio is the same a ____________________________.

_________________________ solids have equal ratios of corresponding linear measures.

## Compare Surface Areas and Volumes of Similar Figures

When you compare similar 2-dimensional figures, area changes as a function of the square of the ratio of corresponding linear measures.

For example, take a look at the areas of these two similar figures:

The ratio between corresponding sides is:

length (A)length (B)=126=21\begin{align*}\frac{length \ (A)}{length \ (B)} = \frac{12}{6} = \frac{2}{1}\end{align*}

The ratio between the areas of the 2 figures is the square of the ratio of the linear measurement:

area (A)area (B)=12864=9624=41 or (21)2\begin{align*}\frac{area \ (A)}{area \ (B)} = \frac{12 \cdot 8}{6 \cdot 4} = \frac{96}{24} = \frac{4}{1} \ \text{or} \ \left (\frac{2}{1}\right )^2\end{align*}

This relationship holds for solid figures as well:

The ratio of the areas of 2 similar figures is equal to the square of the ratio between the corresponding linear sides.

1. True/False: If 2 solid shapes are similar, then the ratios of all of their linear measurements (such as height, width, and depth) are the same.

2. True/False: If 2 solid shapes are similar, then the ratio of their areas is the square of the ratio of their side lengths.

Example 2

Find the ratio of the volume between the two similar figures C and D below.

As with surface area, since the 2 figures are similar you can use the height, depth, or width of the figures to find the linear ratio. In this example we will use the widths of the 2 figures:

width (C)width (D)=1520=34\begin{align*}\frac{width \ (C)}{width \ (D)} = \frac{15}{20} = \frac{3}{4}\end{align*}

The ratio between the volumes of the 2 figures is the cube of the ratio of the linear measurements:

volume (C)volume (D)=(34)3=2764\begin{align*}\frac{volume \ (C)}{volume \ (D)} = \left (\frac{3}{4}\right )^3 = \frac{27}{64}\end{align*}

Does this cube relationship agree with the actual measurements? Compute the volume of each figure and compare:

volume (C)volume (D)=691581220=8101920=2764\begin{align*}\frac{volume \ (C)}{volume \ (D)} =\frac{6 \cdot 9 \cdot 15}{8 \cdot 12 \cdot 20} = \frac{810}{1920} = \frac{27}{64}\end{align*}

As you can see, the ratio holds. We can summarize the information in this lesson here:

If 2 solid figures A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are similar and the ratio of their linear measurements is ab\begin{align*}\frac{a}{b}\end{align*}, then the ratio of their surface areas is:

surface area (A)surface area (B)=(ab)2\begin{align*}\frac{surface \ area \ (A)}{surface \ area \ (B)} = \left (\frac{a}{b}\right )^2\end{align*}

and the ratio of their volumes is:

volume (A)volume (B)=(ab)3\begin{align*}\frac{volume \ (A)}{volume \ (B)} = \left (\frac{a}{b}\right )^3\end{align*}

1. When something is squared, what power is it raised to?

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2. When something is cubed, what power is it raised to?

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3. If 2 similar polyhedra have heights that are in a ratio of 25\begin{align*}\frac{2}{5}\end{align*}, what is the ratio of their surface areas?

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4. If 2 similar polyhedra have depths that are in a ratio of 14\begin{align*}\frac{1}{4}\end{align*}, what is the ratio of their volumes?

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5. True/False: The ratio of side lengths of similar solids is equal to the ratio of their surface areas, and it is also equal to the ratio of their volumes.

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