What is the formula for the area of a regular polygon?

Area = (n times s squared) divided by (4 times tan(PI divided by n)), where n is the number of sides and s is the side length. An equivalent form is (1/2) times perimeter times apothem.

What is the difference between the circumradius and the inradius?

The circumradius R is the distance from the centre to a vertex — the radius of the circle that passes through all corners. The inradius r (also called the apothem) is the distance from the centre to the midpoint of a side — the radius of the inscribed circle. For any regular polygon, R is always larger than r.

How do I find the area if I only know the circumradius?

Select Circumradius from the drop-down and enter R. The tool back-solves the side length as s = 2R times sin(PI divided by n), then computes area from s.

How do I find the area from the inradius or apothem?

Select Inradius / apothem and enter r. The tool inverts r = s divided by (2 times tan(PI divided by n)) to get s = 2r times tan(PI divided by n), then computes area.

Can I find the area from the perimeter alone?

Yes — select Perimeter, enter the total perimeter P, and the tool divides by n to recover the side length, then computes area.

Is there a simpler area formula using the apothem?

Area = (1/2) times perimeter times apothem. This works because each of the n identical triangles from the centre has base s and height r (the apothem), and n times (1/2 times s times r) = (1/2) times ns times r = (1/2) times P times r.

Regular Polygon Area Calculator

A regular polygon has equal sides and equal angles, which means a small set of formulas describes every shape from an equilateral triangle to a regular hectagon and beyond. This calculator accepts any one known measurement — side length, perimeter, circumradius or inradius — and returns every other property including the area, with an SVG diagram and the full working so you can follow every step.

How it works

Given n sides of length s, all properties follow from one set of identities rooted in the fact that the polygon can be divided into n congruent isosceles triangles, each with two sides equal to the circumradius R and a base of length s.

The central formulas are:

Area (A)          = (n * s^2) / (4 * tan(PI / n))
Perimeter (P)     = n * s
Inradius / apothem (r) = s / (2 * tan(PI / n))
Circumradius (R)  = s / (2 * sin(PI / n))
Interior angle    = (n - 2) * 180 / n   [degrees]
Exterior angle    = 360 / n             [degrees]

Solve for variable: if you supply R instead of s, the tool inverts R = s / (2 * sin(PI/n)) to get s = 2 * R * sin(PI/n). If you supply the inradius r, it inverts r = s / (2 * tan(PI/n)) to get s = 2 * r * tan(PI/n). If you supply the perimeter P, it uses s = P / n. In every case the back-solved side length feeds into all other formulas.

Area from the apothem (alternative view): slicing the polygon into n triangles from the centre, each with base s and height r, gives A = n * (1/2 * s * r) = (1/2) * P * r. Both forms yield identical results; the calculator uses the tangent form internally.

Worked example

A regular octagon (n = 8) with side length 7:

Perimeter = 8 x 7 = 56
Interior angle = (8 - 2) x 180 / 8 = 135 degrees
Exterior angle = 360 / 8 = 45 degrees
Inradius = 7 / (2 x tan(22.5 degrees)) = 7 / (2 x 0.41421) approx 8.4497
Circumradius = 7 / (2 x sin(22.5 degrees)) = 7 / (2 x 0.38268) approx 9.1459
Area = (8 x 49) / (4 x tan(22.5 degrees)) approx 236.59

Equivalently: Area = (1/2) x 56 x 8.4497 approx 236.59. Both routes agree.

Sides (n)	Name	Interior angle	Area (s = 10)
3	Equilateral triangle	60 deg	43.30
4	Square	90 deg	100.00
5	Pentagon	108 deg	172.05
6	Hexagon	120 deg	259.81
8	Octagon	135 deg	482.84
12	Dodecagon	150 deg	1119.62

Formula note

The tangent form A = (n * s^2) / (4 * tan(PI/n)) follows directly from summing the n centre triangles: each has area (1/2) * s * r where r = s / (2 * tan(PI/n)), so n * (1/2) * s * s / (2 * tan(PI/n)) = (n * s^2) / (4 * tan(PI/n)). As n grows large the polygon approaches a circle; at n = 1000 with s = 1 the area is approximately PI * R^2, confirming the formula degrades gracefully to the circle formula. The circumradius form R = s / (2 * sin(PI/n)) derives from the law of sines applied to each of the n centre triangles, each subtending an angle of 2 * PI / n at the centre. All calculations run locally in your browser — nothing is sent to any server.