Circumradius

The circumradius of a polygon is the radius of its circumcircle, the circle that passes through all the polygon's vertices. A polygon that has such a circle is called cyclic. All triangles are cyclic, so every triangle has a circumradius; for other polygons existence depends on their angles.

In a triangle with sides a,b,c opposite angles A,B,C, the circumradius R satisfies a = 2R sin A etc., so R = a/(2 sin A) = b/(2 sin B) = c/(2 sin C). Equivalently, using area Δ, R = abc/(4Δ). In a right triangle, R equals half the hypotenuse.

For a regular n-gon with side length s, the circumradius is R = s/(2 sin(π/n)). If the circumradius is known, the side length is s = 2R sin(π/n). In general, the circumcenter is the intersection of the perpendicular bisectors of the sides, and computing R involves fitting a circle through at least three vertices.

Relations with other radii: In triangles, the inradius r and circumradius R are linked by Euler's formula OI^2 = R(R - 2r), where O is the circumcenter and I the incenter. Consequently R ≥ 2r, with equality only in the equilateral case.