Midradius

The radius of the midsphere of a polyhedron, also called the interradius. Let be a point on the original polyhedron and the corresponding point on the dual. Then because and are inverse points, the radii , , and satisfy

(1)

The above figure shows a plane section of a midsphere.

Let be the inradius the dual polyhedron, circumradius of the original polyhedron, and the side length of the original polyhedron. For a regular polyhedron with Schläfli symbol , the dual polyhedron is . Then

			(2)
			(3)
			(4)

Furthermore, let be the angle subtended by the polyhedron edge of an Archimedean solid. Then

			(5)
			(6)
			(7)

so

(8)

(Cundy and Rollett 1989).

For a Platonic or Archimedean solid, the midradius of the solid and dual can be expressed in terms of the circumradius of the solid and inradius of the dual gives

			(9)
			(10)

and these radii obey

(11)