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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 $a, b$ and $c$ denote the triangle's three sides, and let $A$ denote the area of the triangle. Then, the measure of the of the circumradius of the triangle is simply $\frac{abc}{4A}$

Euler's Theorem for a Triangle

Let $\triangle ABC$ have circumradius $R$ and inradius $r$. Let $d$ be the distance between the circumcenter and the incenter. Then we have \[d^2=R(R-2r)\]

See also

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