Polygon Calculator
Calculate area, perimeter, angles, and other properties of regular polygons
Calculate Regular Polygon Properties
Polygon name: Hexagon
Example Calculation
Regular Hexagon Example
Given: Regular hexagon (6 sides) with side length = 4 units
Number of sides (n): 6
Side length (a): 4 units
Step-by-Step Calculation
1. Area: A = 6 × 4² × cot(π/6) / 4 = 6 × 16 × √3 / 4 ≈ 41.57 units²
2. Perimeter: P = 6 × 4 = 24 units
3. Interior angle: α = (6-2) × 180° / 6 = 120°
4. Exterior angle: β = 360° / 6 = 60°
5. Circumradius: R = 4 / (2 × sin(π/6)) = 4 units
6. Inradius: r = 4 / (2 × tan(π/6)) ≈ 3.464 units
Common Polygons
Formula Quick Reference
Area
A = n × a² × cot(π/n) / 4
Perimeter
P = n × a
Interior Angle
α = (n-2) × 180° / n
Exterior Angle
β = 360° / n
Circumradius
R = a / (2 × sin(π/n))
Inradius
r = a / (2 × tan(π/n))
Understanding Regular Polygons
What is a Regular Polygon?
A regular polygon is a 2D closed figure made up of straight line segments where all sides are equal in length and all interior angles are equal in measure. Regular polygons are both equilateral (equal sides) and equiangular (equal angles).
Key Properties
- •Equilateral: All sides have the same length
- •Equiangular: All angles are equal in measure
- •Symmetrical: Has rotational and reflectional symmetry
- •Circumscribed: Can be inscribed in a circle
Important Measurements
Circumradius (R)
The radius of the circle that passes through all vertices of the polygon. Also called the circumscribed circle radius.
Inradius (r) / Apothem
The radius of the largest circle that can fit inside the polygon. Also the perpendicular distance from center to any side.
Interior vs Exterior Angles
Interior angles are inside the polygon. Exterior angles are formed by extending one side. Interior + Exterior = 180°.
Central Angle
The angle formed at the center of the polygon by two radii to adjacent vertices. Always equals 360°/n.