Apothem

The apothem of a regular polygon is the distance from its center to any of its sides. It is the length of the perpendicular segment from the center to a side and equals the radius of the polygon’s incircle (inradius). Because all sides are tangent to the same inscribed circle, the center is equidistant from every side.

For a regular n-gon with side length s, apothem a, and circumradius R, the following relations hold:

- a = s / (2 tan(π/n))

- s = 2 a tan(π/n)

- a = R cos(π/n) and R = a / cos(π/n)

The apothem also relates to area. If the polygon has perimeter P, then area A = (P · a) / 2 = (n s a) / 2. Expressed via the circumradius, A = (n/2) R^2 sin(2π/n).

Regular polygons have an incircle, whose radius is the apothem; non-regular polygons generally do not have a single common apothem. In that broader context, the term apothem is used specifically for the distance from the center to a side in regular polygons where an inscribed circle exists.