Circles inscribed in and circumscribed about regular polygons

Every regular polygon has both an inscribed and a circumscribed circle. The point O where two adjacent angle bisectors meet is equally far from all vertices (centre of the circumscribed circle) and from all sides (centre of the incircle). So both circles share one centre, called the centre of the polygon. The perpendicular from the centre to a side is the apothem of the polygon and equals the incircle radius r. A regular n-gon with side a has area S = ½·n·a·r, because it splits into n equal triangles with base a and height r. The radii are linked by R² = r² + (a/2)².