Bradley is helping design an amusement park. He sketches his designs by using regular polygons. He also uses regular polygons to sketch the tables that he wants to place in the picnic area. He sketches a table that has 6 sides. What is the sum of the interior angles?

In this concept, you will learn how to relate the sides of a polygon to angle measures.

### Relating the Sides of a Polygon to Angles and Diagonals

Polygons can be divided into triangles using diagonals. This becomes very helpful when you try to figure out the sum of the interior angles of a polygon other than a triangle or a quadrilateral.

The sum of the interior angles of a quadrilateral is \begin{align*}360^\circ\end{align*}. You can divide a quadrilateral into two triangles using diagonals. Each triangle is \begin{align*}180^\circ\end{align*}, so the sum of the interior angles of a quadrilateral is \begin{align*}360^\circ\end{align*}.

Here is one **diagonal** in the quadrilateral.

A **diagonal** is a line segment in a polygon that joins two nonconsecutive vertices. Divide up the polygon using diagonals and figure out the sum of the interior angles.

Here is a hexagon that has been divided into triangles by the diagonals. You can see here that there are four triangles formed. If the sum of the interior angles of each triangle is equal to \begin{align*}180^\circ\end{align*}, and you have four triangles, then the sum of the interior angles of a hexagon is:

\begin{align*}4(180) = 720^\circ\end{align*}

You can follow this same procedure with any other polygon.

If you do not have a picture of the polygon, you can use a formula that involves the number of sides in a polygon.

To better understand how this works, let’s look at a table that shows the number of triangles related to the number of sides in a polygon.

The biggest pattern to notice is that the number of triangles is 2 less than the number of sides. If you know that the sum of the interior angles of one triangle is equal to 180 degrees and if you know that there are three triangles in a polygon, then you can multiply the number of triangles by 180 and that will give you the sum of the interior angles.

Here is the formula.

\begin{align*}x =\end{align*} number of sides

\begin{align*}(x - 2)180 =\end{align*} sum of the interior angles

You can take the number of sides and use that as **\begin{align*}x\end{align*}.**

Then solve for the sum of the interior angles.

Let’s look at an example.

What is the sum of the interior angles of a decagon?

A decagon has ten sides. That is your \begin{align*}x\end{align*} measurement. Now let’s use the formula.

\begin{align*}(x - 2)180 & = (10 - 2)180 \\ 8(180) & = 1440^\circ\end{align*}

The answer is that there are \begin{align*}1440^\circ\end{align*} in a decagon.

### Examples

#### Example 1

Earlier, you were given a problem about Bradley and his picnic table designs.

He sketches a table that has 6 sides. What is the sum of the interior angles?

First, write the number of sides that are in the polygon.

6

Next, plug the number of sides in to the formula.

\begin{align*}(6-2)(180)\end{align*}\begin{align*}(5-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

\begin{align*}4(180) = 720\end{align*}

The answer is **\begin{align*}720^o\end{align*}**. The sum of the interior angles is \begin{align*}720^o\end{align*}.

#### Example 2

Calculate the sum of the interior angles of a regular nonagon.

First, write the number of sides that are in a nonagon.

9

Next, plug the number of sides in to the formula.

\begin{align*}(9 - 2)180\end{align*}

Then, solve for the sum of the interior angles.

\begin{align*}(7)(180) = 1260\end{align*}

The answer is \begin{align*}1260^\circ\end{align*}.

Calculate the sums for each example.

#### Example 3

Calculate the sum of the interior angles of a pentagon.

First, write the number of sides that are in a pentagon.

5

Next, plug the number of sides in to the formula.

\begin{align*}(5-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

\begin{align*}(3)(180) = 540\end{align*}

The answer is \begin{align*}540^o\end{align*}.

#### Example 4

Calculate the sum of the interior angles of a triangle.

First, write the number of sides that are in a triangle.

3

Next, plug the number of sides in to the formula.

\begin{align*}(3-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

\begin{align*}1(180) = 180\end{align*}

The answer is \begin{align*}180^o\end{align*}.

#### Example 5

Calculate the sum of the interior angles of an octagon.

First, write the number of sides that are in an octagon.

8

Next, plug the number of sides in to the formula?'

\begin{align*}(8-2)(180)\end{align*}

Then, solve for the sum of the interior angles.

\begin{align*}6(180) = 1080\end{align*}

The answer is \begin{align*}1080^o\end{align*}.

### Review

Look at each image and name the type of polygon pictured.

Name the number of diagonals in each polygon.

Use the formula to name the sum of the interior angles of each polygon.

- Hexagon
- Pentagon
- Decagon
- Heptagon
- Octagon
- Square

### Review (Answers)

To see the Review answers, open this PDF file and look for section 9.14.