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# Area and Perimeter of Similar Polygons

## Ratio of the areas is the square of the scale factor while the ratio of perimeters is the scale factor.

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Area and Perimeter of Similar Polygons

What if you wanted to create a scale drawing using scale factors? This technique takes a small object, like the handprint below, divides it up into smaller squares and then blows up the individual squares. Either trace your hand or stamp it on a piece of paper. Then, divide your hand into 9 squares, like the one to the right, probably 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*} (meaning you 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. After completing this Concept, you'll be able to explain the results of this technique.

### Guidance

Polygons are similar when the corresponding angles are equal and the corresponding sides are in the same proportion. The scale factor for the sides of two similar polygons is the same as the ratio of the perimeters. In fact, the ratio of any part of two similar shapes (diagonals, medians, midsegments, altitudes, etc.) is the same as the scale factor. The ratio of the areas is the square of the scale factor. An easy way to remember this is to think about the units of area, which are always squared. Therefore, you would always square the scale factor to get the ratio of the areas.

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*}.

#### Example A

The two rectangles below are similar. Find the scale factor and the ratio of the perimeters.

The scale factor is \begin{align*}\frac{16}{24}\end{align*}, which reduces to \begin{align*}\frac{2}{3}\end{align*}. The perimeter of the smaller rectangle is 52 units. The perimeter of the larger rectangle is 78 units. The ratio of the perimeters is \begin{align*}\frac{52}{78}=\frac{2}{3}\end{align*}.

#### Example B

Find the area of each rectangle from Example A. Then, find the ratio of the areas.

\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*}.

#### Example C

Find the ratio of the areas of the rhombi below. The rhombi are similar.

There are two ways to approach this problem. One way would be to use the Pythagorean Theorem to find the length of the \begin{align*}3^{rd}\end{align*} side in the triangle and then apply the area formulas and make a ratio. The second, and easier way, would be to find the ratio of the sides and then square that. \begin{align*}\left( \frac{3}{5} \right)^2=\frac{9}{25}\end{align*}

Watch this video for help with the Examples above.

#### Concept Problem Revisited

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

### Vocabulary

Perimeter is the distance around a shape. The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet, inches, centimeters, etc), write “units.” Area is the amount of space inside a figure. Area is measured in square units. Polygons are similar when their corresponding angles are equal and their corresponding sides are in the same proportion. Similar polygons are the same shape but not necessarily the same size.

### Guided Practice

1. 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?

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

3. Using the ratios from #2, find the length of the base of the smaller triangle if the length of the base of the larger triangle is 24 units.

1. 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, we would multiply the area of the smaller trapezoid by the scale factor. However, we would need to flip the scale factor over to be \begin{align*}\frac{16}{9}\end{align*} because we want the larger area. This means we need to multiply by a scale factor that is larger than one. \begin{align*}A=\frac{16}{9} \cdot 81=144 \ cm^2\end{align*}.

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

3. All you would need to do is multiply the scale factor we found in #2 by 24.

\begin{align*}b=\frac{5}{8} \cdot 24=15 \ units\end{align*}

### Practice

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

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

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

1. \begin{align*}\frac{1}{36}\end{align*}
2. \begin{align*}\frac{4}{81}\end{align*}
3. \begin{align*}\frac{49}{9}\end{align*}
4. \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 \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 \begin{align*}2 \sqrt{3}\end{align*}, find the perimeter of the large triangle.
5. Carol drew two equilateral triangles. Each side of one triangle is 2.5 times as long as a side of the other triangle. The perimeter of the smaller triangle is 40 cm. What is the perimeter of the larger triangle?
6. If the area of the smaller triangle is \begin{align*}75 \ cm^2\end{align*}, what is the area of the larger triangle from #13?
7. 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.
8. 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.
9. 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.
10. The ratio of the areas of two right triangles is \begin{align*}\frac{2}{3}\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 19-22 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?
5. The area of one square on a game board is exactly twice the area of another square. Each side of the larger square is 50 mm long. How long is each side of the smaller square?

### Vocabulary Language: English

Perimeter

Perimeter is the distance around a two-dimensional figure.

Proportion

A proportion is an equation that shows two equivalent ratios.

Ratio

A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.

Scale Factor

A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.

Trapezoid

A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.