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.

In enclosing-ball problems, the circumradius of a nonempty bounded set in Euclidean space is the radius of its smallest enclosing closed ball (Arman et al. 2026). This convention can give a different value for a triangle. For an obtuse triangle, the smallest enclosing disk has the longest side as a diameter.

The following table summarizes the circumradii of some nonregular cyclic polygons.

For a triangle with side lengths , , and ,

where is the semiperimeter.

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

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

and

(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

then gives

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

where is the hypotenuse.

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

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

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

and these radii obey

See also

Carnot's Theorem, Circumcircle, Circumsphere, Cyclic Polygon, Cyclic Quadrilateral, Incircle, Inradius, Midradius, Radius

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References

Arman, A.; Bondarenko, A.; Prymak, A.; and Radchenko, D. "Borsuk Number in Four Dimensions Is at Most Seven." 29 Oct 2026. https://arxiv.org/abs/2610.39078.Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., 1888.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Mackay, J. S. "Historical Notes on a Geometrical Theorem and Its Developments [18th Century]." Proc. Edinburgh Math. Soc. 5, 62-78, 1886-1887.

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Circumradius

Cite this as:

Weisstein, Eric W. "Circumradius." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Circumradius.html

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