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# 10.3: Areas of Similar Polygons

Difficulty Level: 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

1. Are two squares similar? Are two rectangles?
2. Find the scale factor of the sides of the similar shapes. Both figures are squares.
3. Find the area of each square.
4. 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, 2 in×2 in\begin{align*}2 \ in \times 2 \ in\end{align*}. Take a larger piece of paper and blow up each square to be 6 in×6 in\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 6 in×6 in\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 1624=23\begin{align*}\frac{16}{24}=\frac{2}{3}\end{align*}.

PsmallPlarge=2(10)+2(16)=52 units=2(15)+2(24)=78 units\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 5278=23\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:

AsmallAlarge=1016=160 units2=1524=360 units2\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 160360=49\begin{align*}\frac{160}{360}=\frac{4}{9}\end{align*}.

The ratio of the sides, or scale factor was 23\begin{align*}\frac{2}{3}\end{align*} and the ratio of the areas is 49\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 mn\begin{align*}\frac{m}{n}\end{align*}, then the ratio of the areas would be (mn)2\begin{align*}\left(\frac{m}{n}\right)^2\end{align*}.

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

(35)2=925\begin{align*}\left(\frac{3}{5}\right)^2=\frac{9}{25}\end{align*}

Example 4: Two trapezoids are similar. If the scale factor is 34\begin{align*}\frac{3}{4}\end{align*} and the area of the smaller trapezoid is 81 cm2\begin{align*}81 \ cm^2\end{align*}, what is the area of the larger trapezoid?

Solution: First, the ratio of the areas would be (34)2=916\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.

916=81A9AA=1296=144 cm2\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 2564\begin{align*}\frac{25}{64}\end{align*}. What is the scale factor?

Solution: The scale factor is 2564=58\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.

58=b248bb=120=15 units\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 18 in×18 in\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.

1. 35\begin{align*}\frac{3}{5}\end{align*}
2. 14\begin{align*}\frac{1}{4}\end{align*}
3. 72\begin{align*}\frac{7}{2}\end{align*}
4. 611\begin{align*}\frac{6}{11}\end{align*}

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

1. 136\begin{align*}\frac{1}{36}\end{align*}
2. 481\begin{align*}\frac{4}{81}\end{align*}
3. 499\begin{align*}\frac{49}{9}\end{align*}
4. 25144\begin{align*}\frac{25}{144}\end{align*}

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

1. What is the ratio of the areas of the large triangle to one of the small triangles?
2. What is the scale factor of large to small triangle?
3. If the area of the large triangle is 20 units2\begin{align*}20 \ units^2\end{align*}, what is the area of a small triangle?
4. If the length of the altitude of a small triangle is 23\begin{align*}2\sqrt{3}\end{align*}, find the perimeter of the large triangle.
5. Find the perimeter of the large square and the blue square.
6. Find the scale factor of the blue square and large square.
7. Find the ratio of their perimeters.
8. Find the area of the blue and large squares.
9. Find the ratio of their areas.
10. Find the length of the diagonals of the blue and large squares. Put them into a ratio. Which ratio is this the same as?
11. Two rectangles are similar with a scale factor of 47\begin{align*}\frac{4}{7}\end{align*}. If the area of the larger rectangle is 294 in2\begin{align*}294 \ in^2\end{align*}, find the area of the smaller rectangle.
12. 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.
13. 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.
14. 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.

1. 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.
2. Find the scale factor.
3. The diagonals in these rhombi are congruent. Find the length of the diagonals and the sides.
4. What type of rhombi are these quadrilaterals?

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

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