Interior Angles of Polygons

Triangles


90° + 60° + 30° = 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 is any shape with 4 sides)


90° + 90° + 90° + 90° = 360°	80° + 100° + 90° + 90° = 360°
A Square adds up to 360°	Let's tilt a line by 10° ... still adds up to 360°!
The Interior Angles of a Quadrilateral add up to 360°

The internal 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!

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

... its internal 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 internal angles add up to 540°)

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

Shape	Sides	Sum of Internal Angles	Shape	Each Angle
			If it is a Regular Polygon..
Triangle	3	180°		60°
Quadrilateral	4	360°		90°
Pentagon	5	540°		108°
Hexagon	6	720°		120°
...	...	..	...	...
Any Polygon	n	(n-2) × 180°		(n-2) × 180° / n