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The circumradius of a cyclic polygon is the radius of the cirumscribed circle of that polygon. For a triangle, it is the measure of the radius of the circle that circumscribes the triangle. Since every triangle is cyclic, every triangle has a circumscribed circle, or a circumcircle.
Formula for a Triangle
Let and denote the triangle's three sides, and let denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply
Euler's Theorem for a Triangle
Let have circumradius and inradius . Let be the distance between the circumcenter and the incenter. Then we have