Polygon Calculator

A regular polygon is defined as a two-dimensional closed shape with equal-length sides and equal interior angles. The mathematics governing these figures relies on trigonometry and the number of sides (n). The perimeter (P) is simply the product of the side length (s) and the number of sides: P = s × n. The total surface area (A) is calculated using the formula A = (n × s² × cot(π/n)) / 4. Additionally, the interior angle (α) and central angle (β) are derived from the vertex count using α = (n - 2)π / n and β = 2π / n respectively. Finally, the circumcircle radius (R) and incircle radius (r)—also known as the apothem—are computed via R = s / (2 × sin(π/n)) and r = s / (2 × tan(π/n)).

Let us walk through a concrete example using a regular hexagon (a polygon with 6 sides) where each side length measures 8 units. First, we find the perimeter by multiplying the number of sides by the side length: P = 6 × 8 = 48 units. Next, we determine the area by applying the formula A = (6 × 8² × cot(π/6)) / 4. Since cot(π/6) equals the square root of 3 (approximately 1.732), the calculation becomes (6 × 64 × 1.732) / 4 = 665.18 / 4 = 166.28 square units. For the interior angle, α = (6 - 2) × 180° / 6 = 120°. Finally, the incircle radius (apothem) is calculated as r = 8 / (2 × tan(π/6)), which yields approximately 6.93 units.

When working with regular polygons, always double-check your angle unit settings—our equations rely on radians for internal computation, though degrees are often preferred for visual interpretation. Keep in mind that as the number of sides approaches infinity, a regular polygon closely approximates a circle, which can help sanity-check your area and perimeter outputs. If you are designing physical structures, remember that the incircle radius (apothem) represents the distance from the center to the midpoint of any side, a vital metric for fabrication tolerances.