Circumradius

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 possess a circumradius.

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

polygon	inradius
3, 4, 5 triangle
30-60-90 triangle
diamond
golden rectangle
golden triangle
isosceles right triangle
isosceles triangle
rectangle
right triangle

For a triangle with side lengths , , and ,

			(1)
			(2)

where is the semiperimeter.

The circumradius of a triangle is connected to other triangle quantities by a number of beautiful relations, including

			(3)
			(4)
			(5)

where is the inradius and is the semiperimeter of the reference triangle (Johnson 1929, pp. 189-191).

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's vertices. Relabeling the diagram for the Soddy circles with polygon vertices , , and and the radii , , and , and using

			(8)
			(9)
			(10)

then gives

(11)

The hypotenuse of a right triangle is a diameter of the triangle's circumcircle, so the circumradius is given by

(12)

where is the hypotenuse.

The circumradius of a cyclic quadrilateral with side lengths , , , and and semiperimeter is given by

(13)

The circumradius of a regular polygon with sides and side length is given by

(14)

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

			(15)
			(16)

and these radii obey

(17)