About

Miscalculating the area of a regular polygon propagates errors into material estimates, structural loads, and CNC toolpaths. This calculator implements the exact trigonometric identity A = n ⋅ s²4 ⋅ tan(π/n) for n-sided equilateral, equiangular polygons. It accepts four input modes - side length s, apothem a, circumradius R, or perimeter P - and derives every remaining geometric property. The tool assumes Euclidean plane geometry and treats the polygon as perfectly regular. Floating-point precision is limited to 10 significant digits. For polygons with n > 100, results converge toward the circumscribed circle and rounding artifacts may appear at the last decimal.

Formulas

The area of a regular n-sided polygon is derived from decomposing it into n congruent isosceles triangles sharing a common vertex at the center.

From side length s:
A = n ⋅ s²4 ⋅ tan(π / n)

From apothem a:
A = n ⋅ a² ⋅ tan(π / n)

From circumradius R:
A = n ⋅ R² ⋅ sin(2π / n)2

From perimeter P:
A = P²4 ⋅ n ⋅ tan(π / n)

Derived quantities:
a = s2 ⋅ tan(π / n) R = s2 ⋅ sin(π / n) P = n ⋅ s

Interior angle: θ = (n − 2) ⋅ 180°n

Where n = number of sides, s = side length, a = apothem (center to midpoint of a side), R = circumradius (center to vertex), P = perimeter, θ = interior angle.

Reference Data

Polygon	n	Interior Angle	Area (s = 1)	as	Rs
Equilateral Triangle	3	60°	0.4330	0.2887	0.5774
Square	4	90°	1.0000	0.5000	0.7071
Pentagon	5	108°	1.7205	0.6882	0.8507
Hexagon	6	120°	2.5981	0.8660	1.0000
Heptagon	7	128.57°	3.6339	1.0383	1.1524
Octagon	8	135°	4.8284	1.2071	1.3066
Nonagon	9	140°	6.1818	1.3737	1.4619
Decagon	10	144°	7.6942	1.5388	1.6180
Hendecagon	11	147.27°	9.3656	1.7028	1.7747
Dodecagon	12	150°	11.1962	1.8660	1.9319
Pentadecagon	15	156°	17.6424	2.3522	2.4049
Icosagon	20	162°	31.5688	3.1569	3.1962
Triacontagon	30	168°	71.4236	4.7631	4.7880
Hectogon	100	176.4°	795.5130	15.9155	15.9204
Circle limit (n→∞)	∞	180°	πR²	R	R

Frequently Asked Questions

As n increases, the regular polygon converges to a circle. At n = 100, the area is within 0.05% of πR². At n = 1000, the difference drops below 0.0005%. This calculator handles n up to 1000 without precision loss in IEEE 754 double-precision arithmetic.

The apothem a equals R ⋅ cos(π / n). For a hexagon (n = 6), a = 0.8660 ⋅ R. For a triangle, a = 0.5 ⋅ R. The ratio a/R approaches 1 as n → ∞.

No. Star polygons (Schläfli symbol {n/k}) are non-convex and require a different winding-number area formula. This tool strictly computes convex regular polygons where n ≥ 3 and n is a positive integer.

A closed-form trigonometric expression generalizes to any n without storing per-polygon constants. The JavaScript Math.tan function uses hardware-accelerated IEEE 754 routines accurate to 15 - 17 significant digits. Lookup tables would limit coverage and introduce interpolation errors for uncommon n values.

Linear unit conversions are exact rational multipliers (e.g., 1 ft = 0.3048 m exactly by definition). Area conversion squares the factor: 1 ft² = 0.09290304 m². No precision is lost beyond the inherent double-precision limit of approximately 15 significant figures.

Hexagonal tiling in architecture and honeycomb panels, octagonal stop-sign fabrication, pentagonal decorative flooring, bolt-head area for torque calculations (hexagonal cross-section), landscape design of fountain basins, and CNC milling of polygonal pockets. In each case, an error in area directly affects material cost and structural integrity.