apothema

An apothema is a geometric term referring to the line segment drawn from the center of a regular polygon to the midpoint of one of its sides. In a regular polygon, all sides and angles are equal, making the apothema a consistent measurement for each side. This line is perpendicular to the side it intersects, effectively forming a right angle at the midpoint.

The apothema is closely related to the polygon’s radius (the distance from the center to a vertex) and its apothem can be calculated if the side length and the number of sides are known. For a regular polygon with *n* sides of length *s*, the apothema (*a*) can be derived using trigonometry. Specifically, it is equal to half the side length divided by the tangent of the central angle (π/*n*), or expressed mathematically as *a = s / (2 tan(π/n))*.

In the context of regular polygons, the apothema is useful for calculating the area. The area (*A*) of a regular polygon can be determined by multiplying half the perimeter (*P*) by the apothema: *A = (P × a) / 2*. This formula arises because the polygon can be divided into *n* congruent isosceles triangles, each with a height equal to the apothema.

The term "apothem" originates from the Greek words *apo* (meaning "from" or "away") and *thema* (meaning "a laying down"), reflecting its role as a perpendicular distance from the center to a side. While primarily used in Euclidean geometry, the concept extends to other areas of mathematics, including the study of regular polyhedrons and tessellations.