Regular Polygon Calculator
Interior angle, perimeter, area, apothem of a regular n-gon.
Overview
The Regular Polygon Calculator computes the interior angle, exterior angle, perimeter, area and apothem of a regular n-gon given one piece of information — usually side length or circumradius. Whether you're tiling a hexagonal floor or studying classical geometry, you get every dimension in one click.
It is built for architects designing patterns, students proving angle-sum theorems, woodworkers sizing parts and game designers laying out hex grids. Each formula falls out of basic trig but they multiply quickly when you need a full polygon profile.
How it works
For a regular n-gon with side length s:
- Interior angle:
(n - 2) * 180° / n. - Exterior angle:
360° / n. - Apothem (inradius):
a = s / (2 * tan(π / n)). - Circumradius:
R = s / (2 * sin(π / n)). - Perimeter:
P = n * s. - Area:
A = (1/2) * P * a = (1/4) * n * s^2 / tan(π / n).
When you supply circumradius or apothem instead of side length, the calculator inverts the appropriate formula. As n grows large the polygon converges to a circle.
Examples
Hexagon, side 10
→ perimeter 60, area ≈ 259.81, apothem ≈ 8.66, interior 120°
Square, side 5 → perimeter 20, area 25, apothem 2.5
Pentagon, side 6
→ perimeter 30, area ≈ 61.94, interior 108°
100-gon, circumradius 1 → area ≈ 3.1395 (close to π)
FAQ
Why does the area formula have tan(π / n)?
The apothem-from-side relation involves tangent of half the central angle, which is π / n. Squaring it into the area formula keeps the unit consistent.
Is the polygon convex?
Regular polygons are always convex. Star polygons are a different family with non-convex shapes.
What's the difference between apothem and circumradius?
The apothem touches the midpoint of each side. The circumradius touches each vertex. For triangles they're very different; as n grows they converge.
Can I work in degrees?
The displayed interior and exterior angles are in degrees, but internal trigonometric calculations use radians. You don't need to convert manually.
Does the calculator handle irregular polygons?
No. For arbitrary polygons, use the shoelace formula on a list of vertex coordinates.