why when we add more side, the sum of the angles go up by 180 degrees

Question

The sum of the angles of a triangle is 180

For a square it’s 360

For a pentagon it’s 540

For a hexagon it’s 720

Why is that?

In: Mathematics

Answer 1

This property is the case for all non-self-intersecting polygons. One way to think about it is with forming triangles, as others have said.  

A different way is to consider “walking” along the perimeter of the polygon. Think about the exterior angles – how much you need to turn – as you make one loop around. (And choose an orientation as “positive”).  

For polygons without intersections, you need to make one full turn (360 degrees). But you have n sides and n vertices (corners). The interior and exterior angle (as “signed” angles, to be specific) at each corner sum to 180 degrees. So the interior angles must sum to (180n – 360) degrees, which gives the sequence.