# Interior Angles of Polygons

### An Interior Angle is an angle inside a shape

Another example:

## Triangles

The Interior Angles of a Triangle add up to 180°

Let’s try a triangle:

90° + 60° + 30° = 180°

It works for this triangle

Now tilt a line by 10°:

80° + 70° + 30° = 180°

It still works!

One angle went **up** by 10°,

and the other went **down** by 10°

## Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

Let’s try a square:

90° + 90° + 90° + 90° = 360°

A Square adds up to 360°

Now tilt a line by 10°:

80° + 100° + 90° + 90° = 360°

It still adds up to 360°

The Interior Angles of a Quadrilateral add up to 360°

### Because there are 2 triangles in a square …

The interior angles in a triangle add up to **180°** …

… and for the square they add up to **360° **…

… because the square can be made from two triangles!

## Pentagon

A pentagon has 5 sides, and can be made from **three triangles**, so you know what …

… its interior angles add up to 3 × 180° =** 540° **

And when it is **regular** (all angles the same), then each angle is 540**°** / 5 = 108**°**

*(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon’s interior angles add up to 540°)*

The Interior Angles of a Pentagon add up to 540°

## The General Rule

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we **add another 180°** to the total:

| | | If it is a ** Regular Polygon ** (all sides are equal, all angles are equal) |

Shape | Sides | Sum of Interior Angles | Shape | Each Angle |
---|

Triangle | 3 | 180**°** | | 60**°** |

Quadrilateral | 4 | 360**°** | | 90**°** |

Pentagon | 5 | 540**°** | | 108**°** |

Hexagon | 6 | 720**°** | | 120**°** |

Heptagon *(or Septagon)* | 7 | 900**°** | | 128.57…**°** |

Octagon | 8 | 1080**°** | | 135**°** |

Nonagon | 9 | 1260**°** | | 140**°** |

… | … | .. | … | … |

Any Polygon | **n** | (**n**-2) × 180**°** | | (**n**-2) × 180**°** / **n** |

So the general rule is:

Sum of Interior Angles = (**n**-2) × 180**°**

Each Angle (of a Regular Polygon) = (**n**-2) × 180**°** / **n**

Perhaps an example will help:

### Example: What about a Regular Decagon (10 sides) ?

Sum of Interior Angles = (**n**-2) × 180**°**

= (**10**-2)×180**° = **8×180° = **1440°**

And it is a Regular Decagon so:

Each interior angle = 1440**°**/10 = **144°**

Note: Interior Angles are sometimes called "Internal Angles"

Interior Angles Exterior Angles Degrees (Angle) 2D Shapes Triangles Quadrilaterals Geometry Index