Areas of Regular Polygons

A regular polygon splits into congruent triangles, one per side, all meeting at the centre. The apothem is the height of each of those triangles — the perpendicular distance from the centre to the middle of a side.

That gives the formula:

A = ½ × apothem × perimeter

It is the triangle area formula applied n times and tidied up.

Do not mix up the apothem with the radius. The apothem reaches the middle of a side; the radius reaches a vertex. The radius is always longer.

When only the side length is given, find the apothem with trigonometry: the central angle for each triangle is 360/n, half of it sits in the right triangle, and the apothem is the adjacent leg.

For composite shapes, break the figure into pieces you know and add or subtract.

A regular n-gon decomposes into n congruent isosceles triangles with apex at the centre, each of base s and height a (the apothem). Total area is n·(½sa) = ½a·(ns) = ½aP where P is the perimeter.

The apothem is the inradius and the distance to a vertex is the circumradius; they satisfy a = R·cos(180°/n) and s = 2R·sin(180°/n). Each central triangle has apex angle 360°/n, and bisecting it produces the right triangle from which the apothem is computed as a = (s/2)/tan(180°/n).

As n → ∞, the apothem approaches the radius and ½aP approaches ½r·2πr = πr², recovering the circle's area — the classical exhaustion argument.

For composite figures, area is additive over non-overlapping regions, so decomposition and subtraction handle any shape built from known pieces. Shaded-region problems are almost always a subtraction, most often a polygon minus an inscribed circle or a sector.