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.

For a triangle with side lengths a, b, and c,


where s=(a+b+c)/2 is the semiperimeter.

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


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

Let d be the distance between incenter I and circumcenter O, d=IO^_. Then




(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 O_1, O_2, and O_3 and the radii r_1, r_2, and r_3, 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 c is the hypotenuse.

The circumradius of a cyclic quadrilateral with side lengths a, b, c, and d and semiperimeter s is given by


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


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


and these radii obey


See also

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

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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: 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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Cite this as:

Weisstein, Eric W. "Circumradius." From MathWorld--A Wolfram Web Resource.

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