Polygon Calculator

Enter the number of sides and one known measurement — side length, inradius (apothem), or circumradius — and the Polygon Calculator works out all key properties of any regular polygon. You get the area, perimeter, inradius, circumradius, interior angle, and exterior angle in one calculation. Also try the calculate Cube.

Results

Area

Perimeter

Side Length (a)

Inradius / Apothem (r)

Circumradius (R)

Interior Angle (α)

Exterior Angle (β)

Sum of Interior Angles

What is a regular polygon?

A regular polygon is a 2D closed figure where all sides are equal in length and all interior angles are equal in measure. Examples include equilateral triangles, squares, regular pentagons, and regular hexagons. This calculator works exclusively with regular polygons. See also our calculate Cylinder Volume (V).

What is the difference between inradius and circumradius?

The inradius (also called the apothem) is the distance from the center of the polygon to the midpoint of any side — it is the radius of the inscribed circle. The circumradius is the distance from the center to any vertex — it is the radius of the circumscribed circle. For any regular polygon, the circumradius is always larger than the inradius.

How is the area of a regular polygon calculated?

The area formula for a regular polygon with n sides and side length a is: A = (n × a² × cot(π/n)) / 4. Equivalently, using the inradius r: A = n × r × a / 2. This calculator derives the side length from whichever known value you provide, then applies these formulas.

How do interior and exterior angles relate to the number of sides?

The interior angle of a regular n-gon is α = (n − 2) × 180° / n. The exterior angle is β = 360° / n. Interior and exterior angles at each vertex always sum to 180°. As the number of sides increases, the interior angle approaches 180° and the shape approaches a circle. You might also find our Arc Length Calculator useful.

Can this calculator work with irregular polygons?

No — this calculator is designed for regular polygons only, where all sides and angles are equal. For irregular polygons you would need to know each individual side length or angle separately. The perimeter of an irregular polygon is simply the sum of all its side lengths.

What is the sum of interior angles of a polygon?

The sum of interior angles of any polygon (regular or irregular) with n sides is (n − 2) × 180°. For example, a triangle has 180°, a quadrilateral has 360°, and a hexagon has 720°. For a regular polygon, each individual interior angle is this total divided by n.

How many sides can this calculator handle?

The calculator supports any regular polygon from 3 sides (equilateral triangle) up to 1000 sides. A polygon with a very large number of sides closely approximates a circle. Common named polygons include the triangle (3), square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), and decagon (10).

What units does the calculator use?

The calculator works with any consistent unit of length — centimetres, metres, inches, feet, etc. The area result will be in square units of whatever length unit you use. Simply interpret the output numbers in the unit you entered for the input measurement.