Area
4u²
Area enclosed by the shape.
Mathematics · Geometry
Regular polygon calculator — area, perimeter and angles
Calculate area, perimeter, apothem, circumradius and angles from side count and side length of a regular polygon.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
Results
Perimeter
8u
Total boundary length.
Apothem
1u
Radius
1.414u
Interior angle
90°
Exterior angle
90°
Results show up to four significant digits for readability; internal calculations retain full precision.
See calculation details
01Perimeter
Formula
P = nsSubstitution
4 × 2Result
8
02Apothem
Formula
aₚ = s/(2tan(π/n))Substitution
2 / (2 × tan(π / 4))Result
1
03Radius
Formula
R = s/(2sin(π/n))Substitution
2 / (2 × sin(π / 4))Result
1.414
04Area
Formula
S = Paₚ/2Substitution
8 × 1 / 2Result
4
05Interior angle
Formula
α = 180°(n − 2)/nSubstitution
180° × (4 − 2) / 4Result
90
06Exterior angle
Formula
β = 360°/nSubstitution
360° / 4Result
90
01Formulas and symbols
Formulas used
S
P = ns
aₚ = s/(2tan(π/n))
S = Paₚ/2
n: number of sides; s: side; aₚ: apothem; P: perimeter; S: area.
R
R = s/(2sin(π/n))
α = 180°(n−2)/n
β = 360°/n
R: circumradius; α: interior angle; β: exterior angle.
| Symbol | Meaning |
|---|---|
| n | Number of sides |
| s | Side length s (u) |
| P | Perimeter |
| S | Area |
| aₚ | Apothem |
| R | Radius |
| α | Interior angle |
| β | Exterior angle |
02Validation example
Reference numerical case
- n = 4 and s = 2 u give perimeter 8 u, area 4 u², apothem 1 u, circumradius √2 u and interior angle 90°.
03How it works
Split the polygon into n congruent triangles from its centre. Each has base s and height equal to the apothem aₚ; the total area is nsaₚ/2. Trigonometry gives the apothem and circumradius.
The circumradius reaches a vertex. The interior angle α and exterior turning angle β add to 180°; all n exterior angles sum to 360°.
04Limits and conventions
A convex regular polygon with an integer 3 ≤ n ≤ 1000 and positive equal sides. All lengths use u. Above 100 sides, the visual becomes a labelled circular approximation; results still use the exact n.
05Common mistakes
These relations do not apply to an irregular polygon. The apothem reaches the midpoint of a side; the circumradius reaches a vertex.
06Practical questions
What is the difference between apothem and circumradius?
The apothem is the perpendicular distance from the centre to a side. The circumradius is the distance from the centre to a vertex, so it is larger than the apothem for a finite regular polygon.
How does the number of sides affect the interior angle?
For a regular n-sided polygon, each interior angle is 180°(n−2)/n. As n increases, the interior angle approaches 180° while the exterior angle 360°/n approaches zero.