Last updated: 2026-05-18

Definition

A regular polygon is a flat shape with equal side lengths and equal interior angles — the equilateral triangle, square, regular pentagon, hexagon, and so on. It is fully determined by two numbers: the number of sides $n$ (an integer with $n \geq 3$) and a single side length $a$. From these, every other measurement follows from three geometric relationships: the polygon divides into $n$ congruent triangles meeting at the centre, the circumscribed circle passes through every vertex, and the inscribed circle touches the midpoint of every side.

This calculator takes the number of sides and the side length and returns the perimeter, area, interior angle, exterior angle, inradius (apothem), and circumradius.

Formulas

The perimeter is the side length repeated $n$ times:

$P = n \cdot a$

The interior angle at each vertex is the total angular sum $(n-2) \times 180°$ divided equally among the $n$ vertices:

$\alpha = \frac{(n-2) \times 180°}{n}$

The exterior angle is the supplement of the interior angle, and the exterior angles always sum to 360° around the polygon:

$\beta = \frac{360°}{n}$

The inradius (apothem) is the perpendicular distance from the centre to any side:

$r = \frac{a}{2 \tan(\pi/n)}$

The circumradius is the distance from the centre to any vertex:

$R = \frac{a}{2 \sin(\pi/n)}$

The area equals half the perimeter times the apothem, which simplifies to:

$A = \frac{n \cdot a^2}{4 \tan(\pi/n)}$

Worked example

A regular hexagonal tile has a side length of 8 cm. For $n = 6$ and $a = 8$ cm, the interior angle is

$\alpha = \frac{(6-2) \times 180°}{6} = 120°$

the apothem is

$r = \frac{8}{2 \tan(30°)} = \frac{8}{2 \times 0.5774} \approx 6.928 \text{ cm}$

the circumradius is

$R = \frac{8}{2 \sin(30°)} = \frac{8}{2 \times 0.5} = 8 \text{ cm}$

and the area is

$A = \frac{6 \times 64}{4 \tan(30°)} = \frac{384}{2.309} \approx 166.3 \text{ cm}^2$

Interior angle by number of sides

As $n$ increases, the interior angle grows toward 180° and the polygon approaches a circle.

Applications

The circumradius of a regular hexagon equals its side length, a consequence of the central triangles being equilateral. This relationship makes hexagonal grids straightforward to lay out, which is one reason hexagonal tilings recur in honeycombs, tiling patterns, and mesh structures. More generally, the apothem is the practical quantity for area, since multiplying it by half the perimeter gives the enclosed surface for any regular polygon.