# 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

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

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

Or, multiply the *number* of triangles by

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

The sum of the **interior angles** in the *hexagon* is

- To find the total measure of _________________________ angles in a polygon, multiply the number of __________________________ you can draw inside the polygon by
180∘ .

**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

So, the sum of the interior angles is

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

- 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

The **sum of the interior angles** of a polygon with

**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

So, the sum of the interior angles in a *nonagon* is

**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.*

2. *Fill in the blanks:*

*The sum of __________________ angles in any polygon is equal to (the number of ___________________ in the polygon minus _________) times*

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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