# Interior Angles of Polygons

## Triangles

The Interior Angles of a Triangle add up to 180°

### 80° + 70° + 30° = 180°

It works for this triangle!

Let's tilt a line by 10° ...

It still works, because one angle went up by 10°, but the other went down by 10°

(A Quadrilateral has 4 straight sides)

### 80° + 100° + 90° + 90° = 360°

A Square adds up to 360°

Let's tilt a line by 10° ... still adds up to 360°!

## Because there are Two Triangles in a Square

 The interior angles in this triangle add up to 180° (90°+45°+45°=180°) ... and for this 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 if it is a regular pentagon (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

Shape Sides Sum of Shape Each Angle Interior Angles If it is a Regular Polygon (all sides are equal, all angles are equal) 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"