Polygon Area Calculator

About This Tool

The Polygon Area Calculator computes the area, perimeter, interior angles, apothem, and circumradius of any regular polygon given the number of sides and the side length. A regular polygon has all sides equal and all interior angles equal, making it one of the most symmetric shapes in Euclidean geometry. This tool is valuable for students learning geometry, architects designing tiled floors and decorative patterns, engineers creating symmetrical structures, and game developers generating procedural shapes.

The Regular Polygon Area Formula

The area of a regular polygon with n sides of length s is A = (n × s²) / (4 × tan(π/n)). This formula works by dividing the polygon into n identical isosceles triangles, each with a base of length s and a height equal to the apothem (the perpendicular distance from the center to the midpoint of a side). Since each triangle has area (1/2) × s × apothem, and there are n triangles, the total area is (n × s × apothem) / 2. Substituting apothem = s / (2 × tan(π/n)) yields the compact formula above. As n increases, the polygon approaches a circle, and the formula converges to πr².

Interior and Exterior Angles

The sum of interior angles of any n-sided polygon is (n - 2) × 180°. For a regular polygon, each interior angle is this sum divided by n: ((n - 2) × 180°) / n. For example, each angle of a regular hexagon is 120°, and each angle of a regular octagon is 135°. The exterior angle at each vertex is the supplement: 360°/n. The exterior angles always sum to 360° regardless of n, a beautiful result that follows from the fact that walking along the perimeter and turning at each vertex completes exactly one full rotation.

Apothem and Circumradius

The apothem is the distance from the center of the polygon to the midpoint of any side. It is essentially the inradius — the radius of the largest circle that fits inside the polygon. The formula is apothem = s / (2 × tan(π/n)). The circumradius is the distance from the center to any vertex — the radius of the smallest circle that contains the polygon. It equals R = s / (2 × sin(π/n)). Together, the apothem and circumradius define the inscribed and circumscribed circles of the polygon. These measurements are critical in mechanical engineering for designing gears (where the pitch circle corresponds to the circumradius) and in architecture for designing columns and rotundas.

Tessellations and Tiling

Only three regular polygons tile the plane by themselves: equilateral triangles (interior angle 60°), squares (90°), and regular hexagons (120°). This is because only these angles evenly divide 360°. The hexagonal tiling is particularly efficient — it covers a plane with the least total perimeter for a given cell area, which is why honeycombs use hexagonal cells. Understanding polygon angles helps architects and designers determine which shapes can tile without gaps. Semi-regular tessellations use combinations of different regular polygons, and determining which combinations work requires knowing the exact interior angles this calculator provides.

From Polygons to Circles

As the number of sides n approaches infinity, a regular polygon converges to a circle. The area formula A = (n × s²) / (4 × tan(π/n)) approaches πr², and the perimeter approaches 2πr. This relationship was used by Archimedes to approximate π by inscribing and circumscribing polygons around a circle and computing their perimeters. With a 96-sided polygon, he determined that π lies between 3.1408 and 3.1429. This polygon-to-circle convergence is also used in computer graphics, where circles are approximated by polygons with enough sides to appear smooth at the rendering resolution.