Polygon Calculator: Your Handy Math Buddy

Formulas for a Regular Polygon

Every property here follows from just two inputs: the number of sides (n) and the side length (s). Perimeter is the simplest: n x s. The rest depend on trigonometry - apothem = s / (2 x tan(pi/n)), area = (perimeter x apothem) / 2, and circumradius = s / (2 x sin(pi/n)).

Worked Example: Regular Hexagon, Side Length 10

Perimeter: 6 x 10 = 60. Apothem: 10 / (2 x tan(30 degrees)) = 10 / (2 x 0.57735) = 10 / 1.1547 = 8.6603. Area: (60 x 8.6603) / 2 = 519.615 / 2 = 259.81 square units. Interior angle: ((6-2) x 180) / 6 = 720/6 = 120 degrees. Exterior angle: 360/6 = 60 degrees. Circumradius: 10 / (2 x sin(30 degrees)) = 10 / 1 = 10.00.

Interior vs. Exterior vs. Central Angles

The interior angle is measured inside the polygon at each vertex; it grows as sides increase (a triangle's interior angle is 60 degrees, an octagon's is 135 degrees). The exterior angle is what's left after turning the corner (180 degrees minus the interior angle) and always equals the central angle - the angle at the shape's center between two adjacent vertices - because both are 360 degrees divided evenly across all n corners.

Apothem vs. Circumradius

The apothem measures from the center to the midpoint of a side (the inradius); the circumradius measures from the center to a vertex (a corner). For any polygon with more than 3 sides, the circumradius is always longer than the apothem, and the gap between them shrinks as the polygon adds sides and approaches a circle.

Scope

This tool only handles regular polygons - equal sides, equal angles. Irregular polygons, where side lengths or angles differ, need a different set of formulas entirely and aren't supported here.