Apothemmi

Apothemmi is a geometric term used for the perpendicular distance from the center of a regular polygon to any of its sides. In a regular polygon, all sides and angles are equal, and there exists an incircle that is tangent to every side; the apothemmi is the radius of this incircle. Because the center is equidistant from all sides, the apothemmi serves as the inradius of the polygon.

Formulas and relationships: If the polygon has n sides and each side has length s, the apothemmi r is r = s / (2 tan(pi/n)). If the polygon has a circumradius R (distance from the center to a vertex), then r = R cos(pi/n). The apothem appears in area calculations: the area A of a regular n-gon is A = (1/2) P r, where P is the perimeter (P = n s). Substituting gives A = (n s r)/2, or, using r = s / (2 tan(pi/n)), A = n r^2 tan(pi/n). Another common form is A = (n/2) R^2 sin(2 pi/n).

Applications and notes: The apothem is used to compute areas of regular polygons, to design and analyze tilings and patterns, and to relate the polygon’s inradius to its side length and number of sides. It is specifically defined for regular polygons, where an incircle exists.