Area of a Regular Polygon Calculator
Enter the number of sides and one known measurement to get the area of any regular polygon. Choose from four calculation methods: side length, apothem (inradius), circumradius, or perimeter plus apothem. The calculator also returns perimeter, interior and exterior angles, apothem, circumradius, and the areas of the inscribed and circumscribed circles.
Formula
Worked example
A regular hexagon with side length 10 m: A = 6 × 10² / (4 × tan(pi/6)) = 600 / (4 × 0.5774) ≈ 259.81 m². Apothem = 10 / (2 × tan(pi/6)) ≈ 8.660 m. Circumradius = 10 / (2 × sin(pi/6)) = 10.000 m. Interior angle = (6-2) × 180 / 6 = 120 degrees.
What is a regular polygon?
A regular polygon is a flat, closed shape in which all sides are equal in length and all interior angles are equal in measure. The simplest regular polygon is the equilateral triangle (3 sides), followed by the square (4 sides), pentagon (5), hexagon (6), and so on up to any number of sides. As the number of sides increases, a regular polygon more and more closely approximates a circle. Every regular polygon can be inscribed in a circle (the circumcircle passes through each vertex) and has an inscribed circle (the incircle touches each side at its midpoint). The distance from the centre to a vertex is the circumradius R, and the distance from the centre perpendicular to a side is the apothem r (also called the inradius).
Four ways to calculate the area
There are four equivalent formulas, each using a different known measurement. When you know the side length a: A = n a² / (4 tan(pi/n)). When you know the apothem r: A = n r² tan(pi/n), or equivalently A = P × r / 2, where P = n × a is the perimeter. When you know the circumradius R: A = n R² sin(2pi/n) / 2. All four give the same result for the same polygon, so use whichever matches the measurement you have. This calculator accepts any one of these and derives all the others automatically.
Interior angles, exterior angles, and sum of angles
Every interior angle of a regular n-gon equals (n - 2) × 180 / n degrees. For a triangle that is 60 degrees, for a square 90 degrees, for a hexagon 120 degrees. The exterior angle (the supplement, formed by extending one side) always equals 360 / n degrees, so the exterior angles of any convex polygon always sum to exactly 360 degrees regardless of the number of sides. The sum of all interior angles equals (n - 2) × 180 degrees: 180 degrees for a triangle, 360 degrees for a quadrilateral, 540 degrees for a pentagon, and so on.
Apothem vs. circumradius
The apothem (inradius) r is the perpendicular distance from the centre to the midpoint of any side. The circumradius R is the distance from the centre to any vertex. For the same polygon, the circumradius is always larger than the apothem except in the limiting case of infinite sides (a circle) where they converge. The relationship is r = R × cos(pi/n). The incircle has radius r and area pi × r². The circumcircle has radius R and area pi × R². The polygon area always falls between these two circle areas, and the ratio polygon area / circumcircle area approaches 1 as n grows.
Properties of common regular polygons (unit side length)
| Polygon | Sides | Interior angle | Area (a=1) | Apothem (a=1) | Circumradius (a=1) |
|---|---|---|---|---|---|
| Equilateral triangle | 3 | 60 deg | 0.4330 | 0.2887 | 0.5774 |
| Square | 4 | 90 deg | 1.0000 | 0.5000 | 0.7071 |
| Pentagon | 5 | 108 deg | 1.7205 | 0.6882 | 0.8507 |
| Hexagon | 6 | 120 deg | 2.5981 | 0.8660 | 1.0000 |
| Heptagon | 7 | 128.57 deg | 3.6339 | 1.0383 | 1.1524 |
| Octagon | 8 | 135 deg | 4.8284 | 1.2071 | 1.3066 |
| Nonagon | 9 | 140 deg | 6.1818 | 1.3737 | 1.4619 |
| Decagon | 10 | 144 deg | 7.6942 | 1.5388 | 1.6180 |
| Dodecagon | 12 | 150 deg | 11.1962 | 1.8660 | 1.9319 |
All values calculated for a side length of 1. Scale by multiplying area by a², lengths by a, where a is your actual side length.
Frequently asked questions
What formula gives the area of a regular polygon?
The most commonly used formula is A = n × a² / (4 × tan(pi/n)), where n is the number of sides and a is the side length. Equivalent forms use the apothem r: A = n × r² × tan(pi/n), or the circumradius R: A = n × R² × sin(2pi/n) / 2. All three give the same result; choose the version that matches the measurement you have.
What is the apothem of a regular polygon?
The apothem (also called the inradius) is the perpendicular distance from the centre of the polygon to the midpoint of any side. It is the radius of the largest circle that fits inside the polygon without crossing a side. For a side length a and n sides, the apothem = a / (2 × tan(pi/n)). It is always shorter than the circumradius, which reaches a vertex instead of a side midpoint.
How do I find the area if I only know the perimeter?
Divide the perimeter by the number of sides to get the side length, then apply the standard formula: A = n × a² / (4 × tan(pi/n)). Alternatively, you can use A = P × r / 2, where r is the apothem. You need either the perimeter alone (plus the number of sides) or the perimeter plus the apothem, because the perimeter alone does not fix the shape when n is unknown.
How does the area change as the number of sides increases?
Keeping the side length fixed, the area grows steadily as n increases, approaching but never quite reaching pi × (a / (2pi))² × pi for very large n. Keeping the circumradius fixed, the area approaches the circumcircle area pi × R² as n grows. The chart in this calculator shows both curves so you can visualize the progression from 3 sides toward a circle.
What is the difference between the incircle and circumcircle?
The incircle (inscribed circle) touches each side of the polygon at its midpoint and has radius equal to the apothem r. The circumcircle (circumscribed circle) passes through each vertex and has radius equal to the circumradius R. For any regular polygon, r < R and the polygon area lies between the two circle areas: pi × r² < polygon area < pi × R². As n increases, both circles and the polygon converge to the same shape.
Can I use this calculator for irregular polygons?
No. This calculator is specifically designed for regular polygons, where all sides and all angles are equal. For irregular polygons (unequal sides or angles), you need the coordinates of each vertex and the shoelace formula, or you need to divide the shape into triangles and sum their individual areas.