# Interior Angle Theorem: Definition & Formula

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• 0:00 Interior Angle and Formula
• 1:31 Each Angle of a…
• 2:18 Finding the Number of Sides
• 2:42 Number of Sides and…
• 3:19 Finding Each Angle
• 3:51 Lesson Summary
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Lesson Transcript
Instructor: Eric Istre

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

This lesson will define what an interior angle is, and it will provide and explain how to use the formula for finding the sum of the interior angles of a polygon. Examples will be provided detailing four of the ways it can be used.

## Interior Angle and Formula

Polygons come in many shapes and sizes. They may have only three sides or they may have many more than that. They can be concave or convex. They may be regular or irregular. Regardless, there is a formula for calculating the sum of all of its interior angles. An interior angle would most easily be defined as any angle inside the boundary of a polygon. It is formed when two sides of a polygon meet at a point.

### The Formula for the Sum of the Interior Angles of a Polygon

The formula for calculating the sum of the interior angles of a polygon is the following:

S = (n - 2)*180

Here n represents the number of sides and S represents the sum of all of the interior angles of the polygon.

### Example 1: Finding the Sum of the Interior Angles

For example, suppose you have an octagon. You can find the sum of the interior angles of that polygon. Since an octagon has eight sides, then it will also have eight angles. If you want to know the sum of the angles inside this figure, use the value n=8 in the formula.

S = (8 - 2)*180; which gives you,

S = (6)*180

S = 1080 degrees

The sum of all of the interior angles of an octagon is 1080 degrees.

## Each Angle of a Regular Polygon

If a polygon is called a regular polygon, then this means that all of its sides are congruent and all of its interior angles are congruent. So, you can find the measure of each angle.

Think back to the octagon used in the previous example. If the octagon is regular, then you can divide the sum by the number of angles to find the measure of each angle. Since there are eight angles and each is the same size, then each must be one-eighth of the total.

The formula for calculating the measure of each angle of a regular polygon is S / n.

Remember that the sum is still 1080 degrees.

So, 1080 / 8 = 135 degrees

The measure of each interior angle of a regular octagon is 135 degrees.

## Finding the Number of Sides

The formula for interior angles can also be used to determine how many sides a polygon has if you know the sum of the angles.

Suppose you have a polygon whose interior angles sum to 540 degrees. This will be the value of S in the formula, and n will be the unknown this time.

540 = (n - 2)*180 Divide both sides by 180.

3 = n - 2 Add 2 to both sides.

5 = n

Since the value of n is five, then the polygon is a pentagon.

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