Apothem Calculator for Regular Polygons

The apothem is the perpendicular distance from the centre of a regular polygon to the midpoint of any side. It is the radius of the largest circle that fits exactly inside the polygon, also called the inradius. The apothem is shorter than the circumradius, which reaches all the way out to a corner.

For a regular polygon with n sides each of length s, the apothem is a = s / (2 * tan(pi / n)), where the angle pi / n is in radians. Equivalently a = (s / 2) * cot(pi / n). This exact trigonometric formula follows from the right triangle formed by the centre, a side midpoint, and a vertex.

The area of any regular polygon is one-half the apothem times the perimeter: A = (1/2) * a * P. This mirrors the triangle area of half base times height, because the polygon is made of n identical triangles each with base s and height equal to the apothem. This calculator reports the area as well.

The apothem is in the same length unit as the side length you enter. If the side is in inches, the apothem is in inches. The area is reported in the corresponding square units. Keep your inputs consistent and the outputs follow directly.

A polygon must have at least three sides, so the smallest case handled is an equilateral triangle. This calculator treats any side count below three as invalid, and the side count must be a whole number because a polygon cannot have a fractional number of sides.