Regular Polygon Calculator

Regular polygon: perimeter, area and apothem

A regular polygon has N equal sides of length L and equal interior angles. The perimeter is P = N · L; the apothem (distance from center to side midpoint) is a = L / (2 · tan(π/N)); the circumradius (center to vertex) is R = L / (2 · sin(π/N)). Area can be computed as A = ½ · P · a or directly A = (N · L²) / (4 · tan(π/N)). The interior angle is (N − 2) · 180° / N — 60° for a triangle, 90° for a square, 108° for a pentagon, 120° for a hexagon, 135° for an octagon. Compass-and-straightedge construction is possible exactly for Gauss polygons: N = 2^k · p₁ · p₂ · … where the pᵢ are distinct Fermat primes (3, 5, 17, 257, 65537). Gauss constructed the 17-gon at age 19 in 1796. Example: a regular hexagon with side 10 has P = 60, a = 10/(2·tan(30°)) ≈ 8.66, A = ½·60·8.66 ≈ 259.81.

Applications: architecture, coins and games

Regular polygons shape architecture (the Pentagon in Washington, octagons in Byzantine tilings), coins (12-sided UK pound, 11-sided Canadian loonie), mosaics and Islamic geometric art, tool design (hexagonal Allen-head screws, octagonal handles), and board games like Catan (hex tiles) and Chinese checkers (hexagram).

FAQ

Why does the hexagon tile perfectly? Its 120° interior angle divides 360° exactly three times, so three hexagons meet at every vertex. Only triangles (6×60°), squares (4×90°), and hexagons (3×120°) can tile the plane alone.

What is the apothem useful for? It lets you compute area as ½·P·a, treating the polygon as N congruent triangles each with base L and height a — handy when you know perimeter but not area directly.

Can any regular polygon be constructed with ruler and compass? No. Only those whose N is a product of a power of 2 and distinct Fermat primes. The heptagon (7 sides) and nonagon (9) are famously impossible.

How does area scale as N grows? For fixed circumradius R, area approaches πR² as N → ∞; for fixed side L, area grows roughly as N²/(4π).