# 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) ?

**n**−2) × 180

**°**

**10**−2) × 180

**°**

**1440°**

And for a Regular Decagon:

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

Note: Interior Angles are sometimes called "Internal Angles"