Apothem: definition, formulas, examples and uses

Key properties and definitions

The apothem has several equivalent descriptions. It is:

• the perpendicular distance from the center of a regular polygon to any side;
• the radius of the circle inscribed in the polygon (the inradius); and
• the altitude of any of the congruent isosceles triangles obtained by joining the center to the polygon's vertices.

These relationships make the apothem especially useful for area calculations and for linking side length, circumradius, and central angles.

Formulas and computation

Let n be the number of sides, s the side length, a the apothem, R the circumradius (distance from center to a vertex), and P the perimeter (P = n·s). Important formulas include:

• a = s / (2 tan(π / n)) — derived from the right triangle formed by half a side, the apothem, and a radius vector;
• a = R · cos(π / n) — relating apothem to the circumradius;
• Area A = (1/2) · a · P = (1/2) · a · n · s — obtained by summing areas of n isosceles triangles with height a.

These formulas allow direct computation of the apothem from side length or circumradius and link the apothem to the polygon's area.

Examples and special cases

Some simple regular polygons give familiar apothems:

• Equilateral triangle (n = 3): a = s / (2√3) = s·√3/6.
• Square (n = 4): a = s / 2, because the center lies at equal distance from each side.
• Regular hexagon (n = 6): a = (√3 / 2)·s, since the polygon decomposes into six equilateral triangles about the center.

These values follow directly from the general formula a = s / (2 tan(π / n)).

History, terminology and related uses

The term "apothem" has long been used in Euclidean geometry to describe the inradius of regular polygons; in classical constructions its value helped compute areas before calculus or coordinate methods were commonplace. In some three-dimensional contexts, the word also appears when describing a regular pyramid: the lateral apothem or slant height is the distance from the base center to the midpoint of a base edge measured along the lateral face. While "inradius" is a closely related synonym, "apothem" emphasizes the perpendicular segment to a side.

Distinctions and practical importance

Only regular polygons have a single well-defined apothem common to all sides; irregular polygons do not generally admit such a congruent interior perpendicular. The apothem is particularly useful in engineering, tiling, and computer graphics when computing areas, designing regular features, or converting between side length and inscribed-circle size. Because the apothem ties together perimeter, area and circumscribed geometry, it remains a fundamental concept in elementary and applied geometry.