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.
| polygon | circumradius |
| 3, 4, 5 triangle |  |
| 30-60-90 triangle |  |
| 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
polyhedron, midradius 
, and edge length of the solid 
as
 |  |  |
(15)
|
 |  |  |
(16)
|
and these radii obey
 |
(17)
|
