Regular Polygon Calculator - Angles, Area, Apothem

The Polygon Calculator handles regular polygons: equal sides and equal angles. Enter the number of sides n and the side length to obtain interior and exterior angles, perimeter, area, apothem and circumradius. The diagram inscribes the shape in a circle when circumradius is shown.

Regular polygons generalize equilateral triangles and squares to n-gons used in tiling and engineering bolts. All formulas on this page assume regularity so one side length fixes the scale.

Calculate a regular polygon

A regular polygon has n equal sides and n equal interior angles. Two inputs, n and side length s, determine the entire figure up to rotation. The engine validates n at least 3 and rejects non-integer side counts for n. The figure draws with a vertex on the standard orientation used in the geometry reference.

Perimeter is n times s. Area uses apothem or circumradius forms equivalent for regular polygons. Results update live as n or s change.

Calculate the interior and exterior angles

Interior angles sum to (n - 2) * 180 degrees for any simple polygon. In a regular polygon each interior angle gets an equal share of that sum. Exterior angles sum to 360 degrees and each equals 360 / n in the regular case.

interior = (n - 2) * 180 / n
exterior = 360 / n

For regular hexagon n = 6:

interior = (6 - 2) * 180 / 6 = 120 deg
exterior = 360 / 6 = 60 deg

Calculate the area and perimeter

Perimeter wraps the boundary with n equal steps of length s. Area can be written as half apothem times perimeter or using the tangent form with n and s alone. Pick the form that matches given data on a problem statement.

P = n * s
A = (1/2) * apothem * P
A = (n * s^2) / (4 * tan(pi / n))

For hexagon n = 6, s = 4:

P = 24
A = (3 * sqrt(3) / 2) * s^2 = (3 * sqrt(3) / 2) * 16 approx 41.57

Calculate the apothem and circumradius

Apothem is the perpendicular distance from center to a side, sometimes labeled inradius of the regular polygon. Circumradius is the distance from center to a vertex. Both scale linearly with s for fixed n.

apothem = s / (2 * tan(pi / n))
circumradius = s / (2 * sin(pi / n))

For n = 6 and s = 4, apothem and circumradius follow from these relations and appear numerically in the panel with the hexagon area approx 41.57.

Solve a regular hexagon with side 4

Take n = 6 and s = 4 as the reference regular polygon. Interior angle hits a clean 120 degrees. Area uses the hexagon shortcut coefficient (3 * sqrt(3) / 2) times s squared. 1. Interior angle

(6 - 2) * 180 / 6 = 120 deg

2. Area

A = (3 * sqrt(3) / 2) * 4^2 = (3 * sqrt(3) / 2) * 16 approx 41.57

3. Perimeter

P = 6 * 4 = 24

Frequently asked questions

Summary

Polygon Calculator returns the measures named in the sections above, with substituted arithmetic for every formula and a labelled figure drawn to the entered dimensions. Worked numbers on this page match the geometry reference fixtures.