The circumradius of a cyclic polygon is a radius of the circle inside which the polygon can be inscribed. Similarly, the circumradius
of a polyhedron is the radius of a circumsphere touching each of the polyhedron's
vertices, if such a sphere exists. Every triangle and every tetrahedron has a circumradius,
but not all polygons or polyhedra do. However, regular polygons and regular polyhedra
posses a circumradius.

The following table summarizes the inradii from some nonregular circumscriptable polygons.

Let
be the distance between incenter and circumcenter , .
Then

(6)

and

(7)

(Mackay 1886-1887; Casey 1888, pp. 74-75). These and many other identities are given in Johnson (1929, pp. 186-190).

This equation can also be expressed in terms of the radii of the three mutually tangent circles centered at the
triangle'svertices. Relabeling
the diagram for the Soddy circles with polygon
vertices ,
, and and the radii , , and , and using