Polygon angles & diagonals

A triangle-based explanation for many-sided shapes.

The idea

Draw diagonals from one vertex of a convex n-sided polygon. You obtain n − 2 triangles, so their angle sums add up to (n − 2) × 180°. In a regular polygon, every interior angle is equal: divide this sum by n. Each exterior turning angle is 360°/n.

Work through an example

A regular hexagon has interior-angle sum 720°, so each of its six angles measures 120°. It has n(n − 3)/2 = 6 × 3/2 = 9 diagonals. Each vertex connects diagonally to three non-neighboring vertices; divide by two because each diagonal was counted twice.

What to watch for

The sum formula applies to simple polygons, but equal individual angles require a regular or equiangular polygon. Do not divide the sum equally for an arbitrary irregular polygon.