# The Interior Angles of a Polygon (Mathematics Lesson)

# The Interior Angles of a Polygon

The interior angles of a polygon are the angles between two sides, inside the shape.# The Sum of the Interior Angles of a Polygon

The sum of the interior angles of a polygon is given by the formula:where

**n**is the number of sides of the polygon.

The formula tells you what the interior angles of a polygon add up to.

# How to Find the Sum of the Interior Angles of a Polygon

**Question:**What is the sum of the interior angles of a triangle?

A triangle has 3 sides, so

**n**= 3.

Using the formula.

The interior angles of a triangle add up to 180°.

Read more about how to find the sum of the interior angles of a polygon

# 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:

# How to Find the Interior Angle of a Regular Polygon

**Question:**What is the interior angle of a regular quadrilateral?

A square is a regular quadrilateral. It has 4 sides, so

**n**= 4.

Using the formula.

Each angle in a square is 90°.

Read more about how to find the interior angle of a regular polygon

# 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° |

Octogon | 1080° | 135° |

Nonagon | 1260° | 140° |

Decagon | 1440° | 144° |

Dodecagon | 1800° | 150° |

##### Curriculum

##### Interactive Test

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##### Note

**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 with two sides.

This shape is made up two triangles, and so the interior angles are 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.

Every time another side is added to a shape, another triangle is added, and another 180° is added to the sum of the interior angles.

Note that the number of triangles is always 2 less than the number of sides.

That is, there are

**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°.

Read 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°.

Read more about how the interior and exterior angle of a polygon add up to 180°