Regular Polygon Properties Calculator
Compute the area, perimeter, apothem, or interior angle of any regular polygon given its number of sides and side length. Great for geometry, tiling design, and architectural planning.
About this calculator
A regular polygon has n equal sides of length s and all interior angles identical. Its perimeter is simply P = n × s. Each interior angle measures α = (n − 2) × 180° / n — derived from the fact that the sum of interior angles in any polygon is (n − 2) × 180°. The apothem a (perpendicular distance from center to a side) is a = s / (2 × tan(π/n)). Area uses the apothem: A = (1/2) × P × a, which simplifies to A = (n × s²) / (4 × tan(π/n)). As n increases, a regular polygon increasingly approximates a circle.
How to use
Take a regular hexagon (n = 6) with side length s = 5 cm. Perimeter: P = 6 × 5 = 30 cm. Interior angle: α = (6 − 2) × 180 / 6 = 4 × 30 = 120°. Apothem: a = 5 / (2 × tan(π/6)) = 5 / (2 × 0.5774) = 5 / 1.1547 ≈ 4.330 cm. Area: A = (6 × 5²) / (4 × tan(π/6)) = 150 / (4 × 0.5774) = 150 / 2.3094 ≈ 64.95 cm². Enter sides = 6, side length = 5, and choose your desired property to see the result instantly.
Frequently asked questions
What is the apothem of a regular polygon and how is it used to find area?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It acts as the 'height' of each triangular slice when you divide the polygon into n congruent triangles from the center. The area of each triangle is (1/2) × s × a, and with n triangles the total area is A = (1/2) × n × s × a = (1/2) × P × a. The formula A = (n × s²) / (4 × tan(π/n)) combines these steps for direct calculation.
How does the interior angle of a regular polygon change as the number of sides increases?
The interior angle formula α = (n − 2) × 180° / n shows that angles grow with n. A triangle (n = 3) has 60° angles, a square (n = 4) has 90°, a hexagon 120°, and a dodecagon (n = 12) has 150°. As n approaches infinity, the interior angle approaches 180°, which corresponds to a circle where the 'sides' are infinitesimally small arcs. This progressive increase is also why regular hexagons tile a flat plane perfectly — their 120° angles sum to exactly 360° at each vertex.
Why do regular hexagons appear so often in nature and engineering?
Regular hexagons are the most efficient shape for tiling a flat surface with equal-area cells using the least total perimeter — a result known as the honeycomb conjecture, proven mathematically in 1999. This efficiency means the least material is needed to enclose the most space, which is why bees naturally construct hexagonal honeycomb cells. In engineering, hexagonal grids appear in materials science (graphene), antenna arrays, and tile flooring because they distribute stress evenly and minimize material waste.