Polygon Calculator

What Is a Regular Polygon?

A polygon is a two-dimensional shape made up of straight line segments. A regular polygon is one where all sides are equal in length and all interior angles are equal. Examples include equilateral triangles, squares, pentagons, hexagons, and octagons.

Our calculator works for any regular polygon with 3 or more sides. Enter the number of sides and any one known value (side length, area, perimeter, circumradius, or apothem) to compute all properties.

Polygon Formulas

Area

A = (n × a²) / (4 × tan(π/n))

Where n is the number of sides and a is the side length.

Example (hexagon, n=6, a=5): A = (6 × 25) / (4 × tan(π/6)) = 150 / 2.309 = 64.952

Perimeter

P = n × a

Simply the number of sides times the side length. Hexagon with a=5: P = 6 × 5 = 30.

Interior & Exterior Angles

Interior angle: α = (n − 2) × 180° / n

Exterior angle: β = 360° / n

The sum of all interior angles is always (n − 2) × 180°. Interior + exterior = 180° for each vertex.

Hexagon: α = (6−2)×180/6 = 120°, β = 360/6 = 60°.

Circumradius (R)

The circumradius is the radius of the circle that passes through all vertices:

R = a / (2 × sin(π/n))

Hexagon with a=5: R = 5 / (2 × sin(π/6)) = 5 / 1 = 5. (For a hexagon, R = a!)

Inradius / Apothem (r)

The apothem is the perpendicular distance from the center to the midpoint of a side (inscribed circle radius):

r = a / (2 × tan(π/n))

Hexagon with a=5: r = 5 / (2 × tan(π/6)) = 5 / 1.155 = 4.330.

Number of Diagonals

Diagonals = n(n − 3) / 2

Hexagon: 6(6−3)/2 = 9 diagonals.

Common Polygon Names

Sides	Name	Interior Angle
3	Triangle	60°
4	Square	90°
5	Pentagon	108°
6	Hexagon	120°
7	Heptagon	128.57°
8	Octagon	135°
9	Nonagon	140°
10	Decagon	144°
12	Dodecagon	150°
20	Icosagon	162°

Frequently Asked Questions

What's the difference between regular and irregular polygons?

A regular polygon has all sides and all angles equal. An irregular polygon has sides or angles of different sizes. Our calculator works for regular polygons only, since irregular polygons need all individual side lengths and angles to solve.

What is the difference between circumradius and apothem?

The circumradius (R) goes from the center to a vertex. The apothem (r) goes from the center to the midpoint of a side. The circumradius is always larger than the apothem.

How do interior and exterior angles relate?

At each vertex, interior + exterior = 180°. The sum of all exterior angles is always 360° for any convex polygon, regardless of the number of sides.

Why does a regular hexagon have R = a?

A regular hexagon can be divided into 6 equilateral triangles. The circumradius equals the side length because each triangle has sides equal to the hexagon's side. This makes hexagons uniquely simple — R = a exactly.

Can this calculator handle very large numbers of sides?

Yes — enter any number ≥ 3. With very large n values (e.g., 100+), the polygon closely approximates a circle, and you'll see the area approach πR² and the angles approach 180°.