## The Sum of the Interior Angles of a Polygon

The sum of the interior angles of a polygon is given by the formula:In this formula,**n**is the number of sides of the polygon. The formula tells you what the interior angles of a polygon add up to.

## Example

Imagine you wanted to find the sum of the interior angles of a triangle. A triangle has 3 sides, so**n**= 3. Using the formula:

Sum of interior angles = (n − 2) × 180°

Sum of interior angles = (3 − 2) × 180°

Sum of interior angles = 1 × 180°

Sum of interior angles = 180°

## The Interior Angles of a Regular Polygon

Regular polygons have equal interior angles. There are as many interior angles as there are sides. To find each interior angle in a regular polygon, divide the sum of the interior angles by the number of sides. The formula for each of the interior angles of a regular polygon is:## Example

Imagine you wanted to find an interior angle of a regular quadrilateral.A square is a regular quadrilateral. It has 4 sides, so

**n**= 4. Using the formula:

Interior angle = (n − 2) × 180° ÷ n

Interior angle = (4 − 2) × 180° ÷ 4

Interior angle = 2 × 180° ÷ 4

Interior angle = 360° ÷ 4

Interior angle = 90°

## The Interior Angles of Different Polygons

Shape | Sum of Interior Angles | Interior Angle of Regular Polygon |
---|---|---|

Triangle | 180° | 60° |

Quadrilateral | 360° | 90° |

Pentagon | 540° | 108° |

Hexagon | 720° | 120° |

Heptagon | 900° | 128.57° |

Octagon | 1080° | 135° |

Nonagon | 1260° | 140° |

Decagon | 1440° | 144° |

Dodecagon | 1800° | 150° |

## What Is a Polygon?

A polygon is a 2-dimensional shape with straight sides.## Why Does the Formula Work?

- The interior angles of a three-sided shape (a triangle) add up to
**180°**.

- To make a four-sided shape (a quadrilateral), replace a side of the triangle with two sides.
This shape is made up two triangles, and so the sum of the interior angles is
**2 × 180° = 360°**.

- A five-sided shape (a pentagon) can be made by adding a third triangle, adding another 180° to the sum of the interior angles, so the sum of the interior angles is
**3 × 180° = 540°**.

**n − 2**triangles for every

**n**triangles. Since there are 180° for every triangle, the sum of the interior angles is

**(n − 2) × 180°**for every

**n**sides.

## Exterior Angles

Polygons have exterior angles as well as interior angles.The sum of the exterior angles of a polygon is 360°. Learn more about the exterior angles of a polygon

## An Interior and Exterior Angle in a Polygon Add Up to 180°

An interior and exterior angle in a polygon add up to 180°.how the interior and exterior angle of a polygon add up to 180°

## You might also like...

geometryunderstanding polygonsfinding the interior angle of a regular polygonfinding the sum of the interior angles of a polygon

#### Help Us Improve Mathematics Monster

- Do you disagree with something on this page?
- Did you spot a typo?

__this form__.

#### Find Us Quicker!

- When using a search engine (e.g., Google, Bing), you will find Mathematics Monster quicker if you add
**#mm**to your search term.

#### Share This Page

If you like Mathematics Monster (or this page in particular), please link to it or share it with others.

If you do, __please tell us__. It helps us a lot!

#### Create a QR Code

Use our handy widget to create a QR code for this page...or any page.