Formula and Method for Regular Polygons
A regular polygon has n equal sides and n equal interior angles, all inscribed symmetrically around a common center. Every measurement — area, perimeter, interior angle, apothem, and circumradius — can be derived from just two inputs: the number of sides (n) and the side length (s). This calculator computes the area as A = (1/4) × n × s² × cot(π/n), the perimeter as P = n × s, the interior angle as (n − 2) × 180° / n, and the apothem as a = s / (2 × tan(π/n)).
How the calculation works
Split a regular n-gon into n identical isosceles triangles that meet at the center. The height of each triangle (from the center to the midpoint of a side) is the apothem, a = s / (2tan(π/n)), and each triangle's area is (1/2) × s × a. Multiplying by n triangles gives the total area, A = (n × s × a) / 2 = (1/4) × n × s² × cot(π/n) — since (1/2)a·s·n = (1/2)·(s/(2tan(π/n)))·s·n. The perimeter is simply the side length times the number of sides, P = n × s. The sum of all interior angles of any simple n-gon is (n − 2) × 180°, so each interior angle of a regular polygon is that total divided evenly by n. The circumradius (center-to-vertex distance) follows a similar triangle relationship: R = s / (2sin(π/n)).
Common mistakes
- Apothem vs. circumradius: the apothem (center to edge midpoint) is always shorter than the circumradius (center to vertex) — do not use one where the other is required.
- Angle units in the formula: the trig functions here use radians (π/n), not degrees; most calculator software handles this internally, but hand calculations must convert n into π/n radians first.
- Mixing side length and diagonal: only the side length s belongs in these formulas — a diagonal or width-across-flats measurement must first be converted to a side length.
Real-world applications
- Architecture and construction use polygon area formulas to size hexagonal or octagonal rooms, gazebos, and paver patios.
- Manufacturing and machining use the interior angle and apothem to cut regular polygon parts (hex nuts, octagonal flanges) to precise tolerances.
- Land surveying and civil engineering use perimeter and area formulas for regular-shaped plots and traffic islands.
- Game design and computer graphics use these formulas to generate and scale regular polygon meshes (stop signs, honeycomb grids, polygon-based sprites).