Polygon Angle Sums

Split a polygon into triangles by drawing every diagonal from one vertex. An n-sided polygon breaks into n − 2 triangles, so its interior angles total:

(n − 2) × 180°

A pentagon gives 3 × 180 = 540°. An octagon gives 6 × 180 = 1080°.

For a *regular* polygon, every angle is equal, so divide by n. Each angle of a regular octagon is 1080 ÷ 8 = 135°.

The exterior angles behave much more simply. They always total 360°, no matter how many sides. Walking once around any convex polygon turns you through one full circle.

So each exterior angle of a regular n-gon is 360/n, and the interior angle is its supplement — often the faster route.

For a convex n-gon, drawing all diagonals from a single vertex produces n − 2 triangles, giving an interior angle sum of (n − 2)·180°. Each interior angle of a regular n-gon is therefore (n − 2)·180°/n.

The exterior angle sum is 360° for every convex polygon, independent of n, because traversing the boundary rotates the direction of travel through exactly one revolution. Each exterior angle of a regular n-gon is 360°/n, and since interior and exterior angles at a vertex form a linear pair, the interior angle is 180° − 360°/n — usually the quicker computation.

Working backwards is a standard question type: given an interior angle, solve (n − 2)·180/n = θ for n, or more simply n = 360/(180 − θ). A non-integer result means no such regular polygon exists.

The exterior formula also explains which regular polygons tile the plane: the interior angle must divide 360° exactly, which happens only for the triangle, square and hexagon.