# 7.7: Change of Dimensions

**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:

\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:

\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:

\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**.

**Reading Check:**

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:

\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:

\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:

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

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

and the *ratio* of their **volumes** is:

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

**Reading Check:**

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

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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

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\begin{align*}{\;}\end{align*}

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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