How to Calculate the Area of a Regular Polygon
This calculator finds the area, perimeter, apothem, and radii of any regular polygon, from equilateral triangles to polygons with 20 sides. Simply select the polygon type and enter the side length.
The Area Formula
The area of a regular polygon with n sides of length l is calculated with: A = (n x l^2) / (4 x tan(pi/n))
This formula is derived by dividing the polygon into n equal isosceles triangles, each with base equal to the polygon's side and height equal to the apothem.
The Apothem and Radii
The apothem is the perpendicular distance from the center to the midpoint of a side. It equals the radius of the inscribed circle: a = l / (2 x tan(pi/n)). The circumscribed circle radius is the distance from the center to each vertex: R = l / (2 x sin(pi/n)).
Interior Angle
The interior angle of a regular polygon depends only on the number of sides: Angle = (n - 2) x 180 / n. For example: equilateral triangle = 60, square = 90, pentagon = 108, hexagon = 120.
Special Cases
- Equilateral triangle (n=3): A = (l^2 x sqrt(3)) / 4
- Square (n=4): A = l^2
- Regular hexagon (n=6): A = (3 x l^2 x sqrt(3)) / 2
As the number of sides increases, the regular polygon increasingly approximates a circle.