# Similar Polygons — Definition, Using & Examples

## What are polygons?

**Polygons** are figures that do all this:

They stay flat in a plane (are two-dimensional)

They enclose a space (no open sides)

They use straight lines (no curves)

A circle, cube, or oval is not a polygon. A triangle is a polygon. The capital letter **L** is not a polygon.

Similar polygons involve two mathematical concepts (similarity and polygons), and have within them an additional concept, proportion.

## Similarity

In mathematics, objects are **similar** if and only if they meet two conditions:

They have congruent corresponding angles

They have proportional corresponding sides

Here are two triangles. Are they similar?

*[insert smaller equilateral △ and larger right △ side by side]*

First look at the angles. Are they the same from one triangle to the other? They are not. Look at the lengths of their sides. An equilateral triangle is not proportionally the same as a right triangle. You could not shrink the right triangle to make it the same size and shape as the smaller equilateral triangle. These figures are *not* similar.

Here are two rectangles. Are they similar?

*[insert 3 x 5 rectangle and 6 x 10 rectangle, using whatever measurement unit you wish]*

The four interior angles of one rectangle are identical to the four interior angles of the other rectangle. Their sides are proportional. The little rectangle could scale up to become the big rectangle. They *are* similar. They have similarity.

## Proportion

A **proportion** is an equality between two ratios. With proportion, you are establishing a ratio of the two measurements of your polygon:

Width

Length

If the ratios between width and length of the two polygons are equivalent, the two shapes are proportional.

Here are two rectangles, one with a width of **6 cm** and length of** 8 cm**, the other with width of **3 cm** and length of** 4 cm**. Are they similar?

*[insert drawing as described]*

Think: Is $\frac{6}{8}$ the same ratio as $\frac{3}{4}$? It is; these two rectangles are similar.

You can also use the combination of interior angles and proportional sides even if you do not have measurements for width and length.

Here are two regular hexagons, one with sides of **3 cm** and the other with sides of **9 cm**. Are they similar?

*[insert drawings as described]*

Think: all the interior angles of both regular hexagons are **120°**, so those fulfill the first condition (congruent corresponding angles). Is **9** a multiple of **3**? In other words, could we enlarge the little hexagon and get the large one? Yes; **9** is a multiple of **3**, so we could enlarge one to make the other. These two hexagons are proportional, so they are similar.

## Try it!

Are these two rectangles similar?

*[insert rectangle labelled 5 x 10 and another 15 x 30 in whatever measurement unit you wish]*

You can compare the measurements of each rectangle to the other: is $\frac{5}{10}$ the same as $\frac{15}{30}$?

You can *also* compare the measures of corresponding parts. Take the short side of one and the short side of the other as a ratio. Take the long sides as the other ratio. Is $\frac{5}{15}$ the same as $\frac{10}{30}$?

These rectangles are similar because their interior angles are congruent and their sides are proportional.

## Transformations

Rotation, reflection, or dilation of one shape does not affect the similarity of two shapes:

*[in a two-by-three grid, insert two similar equilateral triangles, one rotated off axis from the other; second row, two similar right triangles, one a reflection of the other; third row, two similar isosceles triangles, one 50% smaller than the other]*

Whether turned on their heads, flipped over, or greatly enlarged, two figures of the same shape and proportionality are similar.

## Working with similar polygons

You can use your understanding of similar polygons to solve unknowns. Here are two rectangles:

*[insert drawing rectangles 4 x 7 and 12 x 21 labelled in whatever measurement unit you wish, but leave off the "21" on the second rectangle]*

We are told the two rectangles are similar. We do not know the measurement of the unknown side. If they are similar, then they will form equivalent ratios. Set up the relationship as a fraction with an unknown denominator:

$\frac{4}{7}=\frac{12}{x}$

You can solve this by cross-multiplying:

$7\times 12=84$

$84\div 4=21$

The unknown side is **21 units**, since it "scales up" from the other rectangle's side of **7 units**.