A regular polygon has n equal sides and n equal interior angles.
About this calculator
A regular polygon has n equal sides and n equal interior angles. Think of a stop sign (octagon), a hex bolt head, or a pentagonal garden bed. Given n and the side length s, the area formula ns²/(4 tan(π/n)) works for any n ≥ 3.
The apothem is the shortest distance from the center to a side — useful when you tile with hexagons or compute material for a faceted plate. Circumradius is the distance from center to any vertex. Interior angle (n−2)·180°/n tells you the corner you cut when joining panels.
As n grows, the shape approaches a circle. A 6-sided polygon with side 4 has area 41.57 square units; bump n to 12 and the same side length encloses more area. Compare with [[circle]] for the limit, [[area]] for mixed shapes, and [[surface-area]] when extruding the polygon into a prism.
Common uses
- Hexagonal tile or paving layout area estimates
- Octagonal room or gazebo floor area from one wall length
- Engineering bolt-head and nut profile calculations
- School geometry — compare pentagon vs hexagon area at equal side length
- Landscaping beds with equal-sided polygon borders
Frequently asked questions
What is the area of a regular hexagon with side 4?
n = 6, s = 4 gives A = 6×16/(4 tan(30°)) ≈ 41.57 square units. Perimeter is 24 units and the apothem is about 3.46 units.
What is the apothem used for?
It is the inradius to the middle of a side. Area also equals (1/2) × perimeter × apothem, which matches how you would slice the polygon into triangles.
How does a regular polygon relate to a circle?
As n increases with fixed side length, the polygon area approaches the area of a circle whose radius equals the circumradius. At n = 100 the shape is nearly circular.
Can n be less than 3?
No. Three sides is the minimum for a closed polygon in the plane. Enter any integer from 3 upward — hexagon (6) and octagon (8) are common defaults.