$Regular Polygon$

Regular Polygon

Find the side, area, perimeter, radii, interior angle and number of diagonals of a regular polygon from the number of sides and one value.

What is known:

Number of sides and side

Number of sides and circumradius

Number of sides and inradius (apothem)

Number of sides and area

Number of sides n

Side a

Units:

The number of sides is a whole number from 3 to 1000, lengths are positive numbers.

Test yourself

The regular polygon calculator takes the number of sides and one value — the side, the circumradius, the inradius or the area — and finds all elements: side, perimeter, area, both radii, interior and central angles, the sum of the angles and the number of diagonals.

Formulas of a regular n-gon

Value	Formula
Circumradius	R = a / (2 sin(180°/n))
Inradius (apothem)	r = a / (2 tan(180°/n))
Perimeter	P = n · a
Area	S = P · r / 2 = n · a² / (4 tan(180°/n))
Interior angle	α = 180° · (n − 2) / n
Central angle	β = 360° / n
Sum of the angles	180° · (n − 2)
Number of diagonals	N = n(n − 3) / 2

Example. A regular hexagon with side 4 cm.
R = 4 (for a hexagon the circumradius equals the side), r = 2√3 ≈ 3.46 cm
P = 24 cm, S = 24 · 3.46 / 2 ≈ 41.57 cm², α = 120°, 9 diagonals.