# 6.9: Sum of the Interior Angles of a Polygon

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

## Learning Objectives

- Identify the interior angles of convex polygons.
- Find the sums of interior angles in convex polygons.

## Interior Angles in Convex Polygons

The **interior angles** are the angles on the *inside* of a polygon:

As you can see in the image, a polygon has the *same* number of **interior angles** as it does *sides*.

- This means that if a polygon has 3 sides, then it is called a __________________ and it has ______ interior angles.
- If a polygon has 6 sides, then it is called a ________________________ and it has ______ interior angles.
- If a polygon has 8 sides, then it is called a ________________________ and it has ______ interior angles.

## Summing Interior Angles in Convex Polygons

You have already learned the **Triangle Sum Theorem**. It states that the *sum* of the measures of the **interior angles** in a triangle will always be \begin{align*}180^\circ\end{align*}. What about other polygons? Do they have a similar rule?

We can use the **Triangle Sum Theorem** to find the sum of the measures of the angles for *any* polygon. The first step is to *cut the polygon into triangles by drawing* *diagonals**from one* ** vertex**. When doing this you must make sure none of the triangles overlap. Check out the diagram on the next page.

This shape below is a *hexagon* because it has _______ sides.

The *hexagon* above is divided into 4 triangles.

You can see that we have picked a single **vertex** (or *corner*) of the hexagon and drawn a line to each vertex *across* the hexagon. You cannot draw a diagonal to the two vertices next to the starting point because those would be sides.

Since each *triangle* has internal angles that sum to \begin{align*}180^\circ\end{align*}, you can find the sum of the interior angles in the hexagon by adding the triangle angles!

The measure of each angle in the hexagon is a *sum of angles from the triangles*.

Since none of the triangles overlap, we can obtain the TOTAL measure of **interior angles** in the hexagon by summing all of the triangles' interior angles. There are 4 triangles, so add \begin{align*}180^\circ\end{align*} 4 times:

\begin{align*}180^\circ + 180^\circ + 180^\circ + 180^\circ = 720^\circ\end{align*}

Or, multiply the *number* of triangles by \begin{align*}180^\circ\end{align*}:

There are ______ triangles in a *hexagon*, so this is:

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

The sum of the **interior angles** in the *hexagon* is \begin{align*}720^\circ\end{align*}.

- To find the total measure of _________________________ angles in a polygon, multiply the number of __________________________ you can draw inside the polygon by \begin{align*}180^\circ\end{align*}.

**Reading Check**

1. *True or false: Interior angles are the angles inside a polygon*.

2. *True or false: A polygon has the same number of interior angles as it has sides*.

**Example 1**

*What is the sum of the interior angles in the polygon below?*

The shape in the diagram is an *octagon* because it has 8 sides. Draw triangles on the interior using the same process:

The *octagon* can be divided into six triangles.

So, the sum of the **interior angles** will be equal to the sum of the angles in the six triangles (and each triangle is \begin{align*}180^\circ\end{align*}):

\begin{align*}6 \cdot 180^\circ = 1080^\circ\end{align*}

So, the sum of the interior angles is \begin{align*}1080^\circ\end{align*}.

What you may have noticed from these examples is that for *any* polygon, the number of triangles you can draw will be *2 less than the number of sides* (or the number of vertices).

This means that if a polygon has 5 sides, then you can draw \begin{align*}(5 - 2)\end{align*} or 3 triangles.

- Or, if a polygon has 7 sides, then you can draw ( _____ – 2) or _____ triangles.
- Or, if a polygon has 10 sides, then it is called a ____________________ and you can draw ( _____ – 2) or _____ triangles.

You can create an expression for the sum of the interior angles of any polygon using \begin{align*}n\end{align*} for the number of sides of the polygon:

The **sum of the interior angles** of a polygon with \begin{align*}n\end{align*} sides is:

\begin{align*}\text{Angle Sum} = 180^\circ (n - 2)\end{align*}

**Example 2**

*What is the sum of the interior angles of a nonagon?*

To find the sum of the interior angles in a *nonagon*, use the expression on the previous page. Remember that a *nonagon* has 9 sides, so \begin{align*}n\end{align*} will be equal to 9.

\begin{align*}\text{Angle Sum} & = 180^\circ (n - 2)\\ & = 180^\circ (9 - 2)\\ & = 180^\circ (7)\\ & = 1260^\circ\end{align*}

So, the sum of the interior angles in a *nonagon* is \begin{align*}1260^\circ\end{align*}.

**Reading Check**

1. *What is the relationship between the number of sides in a polygon and the number of triangles you can draw inside it with diagonals? Describe in your own words.*

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2. *Fill in the blanks:*

*The sum of __________________ angles in any polygon is equal to (the number of ___________________ in the polygon minus _________) times* \begin{align*}180^\circ\end{align*}.

3. *Challenge:*

*A* *regular**polygon has congruent sides and congruent angles. If you know how to find the TOTAL measure of interior angles in a polygon, how would you find the measure of ONE interior angle in a regular polygon? Describe.*

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