apotema

An apothem of a regular polygon is the perpendicular distance from the center to any of its sides. It equals the radius of the polygon's inscribed circle (inradius). Because a regular polygon has rotational symmetry, all sides are equally distant from the center.

If the polygon has n sides and side length s, the apothem a is a = s / (2 tan(π/n)). If denote R as the circumradius (distance from center to a vertex), then a = R cos(π/n). The polygon's area can be computed as A = (1/2) P a = (1/2) n s a, where P is the perimeter. Substituting a yields A = n a^2 tan(π/n) or, equivalently, A = (1/2) n R^2 sin(2π/n).

In non-regular polygons there is in general no single apothem; the concept is specifically tied to regular polygons and the incircle. For tangential polygons (those with an incircle) a common inradius exists, but the center need not be at the geometric center unless the polygon is regular.