# 10.3: Areas of Similar Polygons

**At Grade**Created by: CK-12

## Learning Objectives

- Understand the relationship between the scale factor of similar polygons and their areas.
- Apply scale factors to solve problems about areas of similar polygons.

## Review Queue

- Are two squares similar? Are two rectangles?
- Find the scale factor of the sides of the similar shapes. Both figures are squares.
- Find the area of each square.
- Find the ratio of the smaller square’s area to the larger square’s area. Reduce it.

**Know What?** One use of scale factors and areas is scale drawings. This technique takes a small object, like the handprint to the right, divides it up into smaller squares and then blows up the individual squares. In this Know What? you are going to make a scale drawing of your own hand. Trace your hand on a piece of paper. Then, divide your hand into 9 squares, like the one to the right, \begin{align*}2 \ in \times 2 \ in\end{align*}. Take a larger piece of paper and blow up each square to be \begin{align*}6 \ in \times 6 \ in\end{align*} (you will need at least an 18 in square piece of paper). Once you have your \begin{align*}6 \ in \times 6 \ in\end{align*} squares drawn, use the proportions and area to draw in your enlarged handprint.

## Areas of Similar Polygons

In Chapter 7, we learned about similar polygons. Polygons are similar when the corresponding angles are equal and the corresponding sides are in the same proportion.

**Example 1:** The two rectangles below are similar. Find the scale factor and the ratio of the perimeters.

**Solution:** The scale factor is \begin{align*}\frac{16}{24}=\frac{2}{3}\end{align*}.

\begin{align*}P_{small} &= 2(10)+2(16)=52 \ units\\ P_{large} &= 2(15)+2(24)=78 \ units\end{align*}

The ratio of the perimeters is \begin{align*}\frac{52}{78}=\frac{2}{3}\end{align*}.

** The ratio of the perimeters is the same as the scale factor.** In fact, the ratio of any part of two similar shapes (diagonals, medians, midsegments, altitudes, etc.) is the same as the scale factor.

**Example 2:** Find the area of each rectangle from Example 1. Then, find the ratio of the areas.

**Solution:**

\begin{align*}A_{small} &= 10 \cdot 16=160 \ units^2\\ A_{large} &= 15 \cdot 24=360 \ units^2\end{align*}

The ratio of the areas would be \begin{align*}\frac{160}{360}=\frac{4}{9}\end{align*}.

The ratio of the sides, or scale factor was \begin{align*}\frac{2}{3}\end{align*} and the ratio of the areas is \begin{align*}\frac{4}{9}\end{align*}. Notice that the ratio of the areas is the ** square** of the scale factor.

**Area of Similar Polygons Theorem:** If the scale factor of the sides of two similar polygons is \begin{align*}\frac{m}{n}\end{align*}, then the ratio of the areas would be \begin{align*}\left(\frac{m}{n}\right)^2\end{align*}.

If the scale factor is \begin{align*}\frac{m}{n}\end{align*}, then the ratio of the areas is \begin{align*}\left(\frac{m}{n}\right)^2\end{align*}.

**Example 3:** Find the ratio of the areas of the rhombi below. The rhombi are similar.

**Solution:** Find the ratio of the sides and square it.

\begin{align*}\left(\frac{3}{5}\right)^2=\frac{9}{25}\end{align*}

**Example 4:** Two trapezoids are similar. If the scale factor is \begin{align*}\frac{3}{4}\end{align*} and the area of the smaller trapezoid is \begin{align*}81 \ cm^2\end{align*}, what is the area of the larger trapezoid?

**Solution:** First, the ratio of the areas would be \begin{align*}\left(\frac{3}{4}\right)^2=\frac{9}{16}\end{align*}. Now, we need the area of the larger trapezoid. To find this, set up a proportion using the area ratio.

\begin{align*}\frac{9}{16} = \frac{81}{A} \rightarrow 9A &= 1296\\ A &= 144 \ cm^2\end{align*}

**Example 5:** Two triangles are similar. The ratio of the areas is \begin{align*}\frac{25}{64}\end{align*}. What is the scale factor?

**Solution:** The scale factor is \begin{align*}\sqrt{\frac{25}{64}}=\frac{5}{8}\end{align*}.

**Example 6:** Using the ratios from Example 5, find the length of the base of the smaller triangle if the length of the base of the larger triangle is 24 units.

**Solution:** Set up a proportion using the scale factor.

\begin{align*}\frac{5}{8} = \frac{b}{24} \rightarrow 8b &= 120\\ b &= 15 \ units\end{align*}

**Know What? Revisited** You should end up with an \begin{align*}18 \ in \times 18 \ in\end{align*} drawing of your handprint.

## Review Questions

- Questions 1-4 are similar to Example 3.
- Questions 5-8 are similar to Example 5.
- Questions 9-18 are similar to Examples 1-3, and 5.
- Questions 19-22 are similar to Examples 4 and 6.
- Questions 23-26 are similar to Examples 5 and 6.

Determine the ratio of the areas, given the ratio of the sides of a polygon.

- \begin{align*}\frac{3}{5}\end{align*}
- \begin{align*}\frac{1}{4}\end{align*}
- \begin{align*}\frac{7}{2}\end{align*}
- \begin{align*}\frac{6}{11}\end{align*}

Determine the ratio of the sides of a polygon, given the ratio of the areas.

- \begin{align*}\frac{1}{36}\end{align*}
- \begin{align*}\frac{4}{81}\end{align*}
- \begin{align*}\frac{49}{9}\end{align*}
- \begin{align*}\frac{25}{144}\end{align*}

This is an equilateral triangle made up of 4 congruent equilateral triangles.

- What is the ratio of the areas of the large triangle to one of the small triangles?
- What is the scale factor of large to small triangle?
- If the area of the large triangle is \begin{align*}20 \ units^2\end{align*}, what is the area of a small triangle?
- If the length of the altitude of a small triangle is \begin{align*}2\sqrt{3}\end{align*}, find the perimeter of the large triangle.
- Find the perimeter of the large square and the blue square.
- Find the scale factor of the blue square and large square.
- Find the ratio of their perimeters.
- Find the area of the blue and large squares.
- Find the ratio of their areas.
- Find the length of the diagonals of the blue and large squares. Put them into a ratio. Which ratio is this the same as?
- Two rectangles are similar with a scale factor of \begin{align*}\frac{4}{7}\end{align*}. If the area of the larger rectangle is \begin{align*}294 \ in^2\end{align*}, find the area of the smaller rectangle.
- Two triangles are similar with a scale factor of \begin{align*}\frac{1}{3}\end{align*}. If the area of the smaller triangle is \begin{align*}22 \ ft^2\end{align*}, find the area of the larger triangle.
- The ratio of the areas of two similar squares is \begin{align*}\frac{16}{81}\end{align*}. If the length of a side of the smaller square is 24 units, find the length of a side in the larger square.
- The ratio of the areas of two right triangles is \begin{align*}\frac{4}{9}\end{align*}. If the length of the hypotenuse of the larger triangle is 48 units, find the length of the smaller triangle’s hypotenuse.

Questions 23-26 build off of each other. You may assume the problems are connected.

- Two similar rhombi have areas of \begin{align*}72 \ units^2\end{align*} and \begin{align*}162 \ units^2\end{align*}. Find the ratio of the areas.
- Find the scale factor.
- The diagonals in these rhombi are congruent. Find the length of the diagonals and the sides.
- What type of rhombi are these quadrilaterals?

## Review Queue Answers

- Two squares are always similar. Two rectangles can be similar as long as the sides are in the same proportion.
- \begin{align*}\frac{10}{25} = \frac{2}{5}\end{align*}
- \begin{align*}A_{small} = 100, A_{large}=625\end{align*}

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