# 6.9: Sum of the Interior Angles of a Polygon

Difficulty Level: 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*}.

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*}.

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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