Regular Polygons

A regular polygon has equal sides and equal angles. Because it is composed of n identical isosceles triangles from the center, its area is simply (perimeter × apothem) ÷ 2, and the side-apothem-radius right triangle determines everything else.

Angles and Diagonals

Interior angles sum to (n − 2) × 180°, so each interior angle is that sum divided by n. The exterior angle is always 360° ÷ n, and the number of diagonals is n(n − 3) ÷ 2 — a pentagon has 5, a hexagon 9, an octagon 20.

Approaching the Circle

As the number of sides grows, a regular polygon converges to its circumscribed circle — which is how the ancient Greeks estimated π by inscribing polygons with hundreds of sides. A 96-gon got Archimedes within 0.03% of π.