the apothem
/ AP-uh-them /
Draw a regular hexagon — a stop-sign-like shape with all sides and angles equal — and mark its centre. The straight distance from that centre to the midpoint of any one side, hitting the side at a right angle, is the apothem. It is the radius of the inscribed circle that just kisses every side from inside.
The apothem is the key that turns a regular polygon into a pile of identical triangles. Connect the centre to all the vertices: a regular polygon with n sides splits into n congruent triangles, each with base s (a side) and height equal to the apothem a. Each triangle has area (1/2) · s · a, so the whole polygon has area n · (1/2) · s · a = (1/2) · (n · s) · a = (1/2) · P · a, where P = n · s is the perimeter. In short: area = half the perimeter times the apothem.
This formula is also a doorway to the circle. As the number of sides n grows, a regular polygon hugs its circumscribed circle ever more closely; the perimeter P approaches the circumference 2 · pi · r and the apothem a approaches the radius r, so (1/2) · P · a approaches (1/2) · (2 · pi · r) · r = pi · r^2 — the area of the circle. The circle is the polygon with infinitely many sides.
A regular hexagon with side 4 has perimeter 24 and apothem 2·sqrt(3) ≈ 3.46, so its area is (1/2)(24)(2·sqrt(3)) = 24·sqrt(3) ≈ 41.6 square units.
Area = (1/2) × perimeter × apothem, the polygon's bridge to pi·r^2.
The apothem is defined only for regular polygons (where every side is the same distance from the centre); an irregular polygon has no single apothem.