Formula and Method for the Area of a Regular Polygon
A regular polygon is a shape with n equal sides and n equal interior angles — an equilateral triangle (n = 3), a square (n = 4), a regular pentagon (n = 5), a regular hexagon (n = 6), and so on. Splitting a regular n-gon into n identical isosceles triangles from its center gives the area formula A = (n × s²) / (4 × tan(π/n)), where s is the side length. This calculator also derives the perimeter, apothem (inradius), and interior angle from the same two inputs.
How the calculation works
Enter the number of sides n and the side length s, then choose the unit. Each of the n triangles formed by connecting the center to two adjacent vertices has a base of s and a height equal to the apothem a = s / (2 × tan(π/n)) — the distance from the center to the midpoint of a side. The area of one triangle is ½ × s × a, and multiplying by n gives A = ½ × (n × s) × a = ½ × perimeter × apothem, which is algebraically equivalent to A = (n × s²) / (4 × tan(π/n)). The perimeter is simply P = n × s, and each interior angle equals (n − 2) × 180° / n.
Common mistakes
- Confusing side length with apothem or circumradius: the apothem (center to edge midpoint) is shorter than the circumradius (center to vertex) — plugging the wrong one into a formula gives the wrong area.
- Radians vs. degrees: the formula A = (n × s²) / (4 × tan(π/n)) uses π/n in radians. If your calculator is in degree mode, use 180/n instead of π/n inside the tangent.
- Non-integer or too-small n: a polygon must have a whole number of sides, and at least 3 — values like n = 2.5 or n = 1 have no valid regular-polygon area.
Real-world applications
- Architecture and construction use polygon area for hexagonal or octagonal floor tiles, gazebos, and paver patterns.
- Manufacturing and machining use it to size hex bolts, nuts, and polygonal gaskets from a known side length.
- Land surveying and civil engineering use it for regularly shaped plots, roundabouts, and traffic islands.
- Education uses the center-triangle decomposition to teach trigonometry, symmetry, and the link between apothem and area.