### Interior Angle Sum Theorem

#### What is true about the sum of angles inside a polygon (ie interior angles)?

The sum of the measures of the interior angles of a convex polygon with *n* sides is
$ (n-2)180^{\circ} $

**Examples:**

- Triangle or ( '3-gon')
- sum of interior angles: $$ (\red 3-2) 180 = 180^{\circ} $$

- Quadrilateral which has four sides ( ' 4-gon')
- sum of interior angles: $$ (\red 4-2) 180 = 360^{\circ} $$

- Hexagon which has six sides ( '6-gon')
- sum of interior angles: $$ (\red 6-2) 180 = 720^{\circ} $$

**Video** Tutorial

on Interior Angles of a Polygon

**Definition of a Regular Polygon:** A regular polygon is simply a polygon whose sides all have the same length and whose angles all have the same measure. The most well known example of a regular polygon is the equilateral triangle

### Measure of a Single Interior Angle

#### What about when you just want 1 interior angle?

In order to find the measure of a single interior angle of a regular polygon (a polygon with sides of equal length and angles of equal measure) with n sides, we just divide the sum of the interior angles or (n-2) × 180 by the number of sides or n

**The Formula**

An interior angle of a **regular** polygon with n sides is
$ \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} $

**Example:**

To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $

180°

Use Interior Angle Theorem:$$ (\red 5 -2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$