Interior Angles of Polygons

An Interior Angle is an angle inside a shape

interior exterior angles

Triangles

The Interior Angles of a Triangle add up to 180°

Let's try a triangle:
interior angles triangle 90 60 30
90° + 60° + 30° = 180°

It works for this triangle


Now tilt a line by 10°:
interior angles triangle 80 70 30
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:
interior angles square 90 90 90 90
90° + 90° + 90° + 90° = 360°

A Square adds up to 360°


Now tilt a line by 10°:
interior angles 100 90 90 80
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 ...

interior angles 90 (45,45) 90 (45,45)

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

interior angles 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° triangle 60°
Quadrilateral 4 360° Quadrilateral 90°
Pentagon 5 540° Pentagon 108°
Hexagon 6 720° Hexagon 120°
Heptagon (or Septagon) 7 900° regular heptagon 128.57...°
Octagon 8 1080° regular octagon 135°
Nonagon 9 1260° regular nonagon 140°
... ... .. ... ...
Any Polygon n (n-2) × 180° regular n gon (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) ?

decagon

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"