Regular polygon calculator
This regular polygon calculator handles any number of sides from the side length, circumradius, inradius or area. For a hexagon with a = 5 the area is 64.95 and the perimeter 30, with the full calculation steps.
Example: Number of sides n 6 · Known value: Side length a · Side length a 5 → Result: Area A = 64.95191 · Circumference C = 30. Source: Wikipedia – Regelmäßiges Polygon. Updated: .
How it is calculated
A regular polygon (n-gon) has n equal sides and n equal angles. It has a circumscribed circle through all vertices (radius R) and an inscribed circle touching all sides (radius ρ). All measures depend only on n and one length.
The formulas
- Circumradius: R = a ÷ (2 · sin(π ÷ n))
- Inradius: ρ = a ÷ (2 · tan(π ÷ n))
- Area: A = n · a · ρ ÷ 2 = n · a² ÷ (4 · tan(π ÷ n))
- Perimeter: P = n · a
- Interior angle: (n − 2) · 180° ÷ n, diagonals: n · (n − 3) ÷ 2
How to use it
Enter the number of sides and choose which measure you know. Example: a hexagon (n = 6) with a = 5 has R = 5, ρ = 4.33, A = 6 · 5 · 4.33 ÷ 2 = 64.95, P = 30, interior angles of 120° and 9 diagonals. In a hexagon the circumradius equals the side length. The unit is up to you; the area is in the square of the chosen unit.
Frequently asked questions
How do you calculate the area of a regular polygon?
A = n · a² ÷ (4 · tan(π ÷ n)). For a hexagon with a = 5 that is 64.95; for a square with a = 2 exactly 4.
How do you calculate the interior angle?
The interior angle is (n − 2) · 180° ÷ n. A pentagon has 108°, a hexagon 120°.
How many diagonals does an n-gon have?
n · (n − 3) ÷ 2. A hexagon has 9, an octagon 20 diagonals.
What is the difference between circumcircle and incircle?
The circumcircle passes through all vertices, the incircle touches every side at its midpoint. The inradius ρ is the distance from the centre to a side midpoint.
How do you find the side length from the circumradius?
a = 2 · R · sin(π ÷ n). Choose “Circumradius” as the known value.
Sources and legal basis
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