Polygon Inscribing

Polygon inscribing is the iterative construction in which a regular polygon is inscribed in a circle and a new circle is inscribed in that regular polygon, with the number of sides increasing from 3 to 4, 5, ... at successive steps. The equation relating the inradius and circumradius of a regular polygon with sides,

(1)

gives the limiting ratio of the radii of the circles as

(2)

Numerically,

(3)

(OEIS A085365), where is the corresponding constant for polygon circumscribing. The constant is termed the Kepler-Bouwkamp constant by Finch (2003). Kasner and Newman's (1989) approximate value is incorrect, as is the value of 0.8700... given by Prudnikov et al. (1986, p. 757).

Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these triangles is the same independent of the polygon vertex chosen (Johnson 1929, p. 193).