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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 circumradius of the triangle is simply . Also,
We let , , , , and . We know that is a right angle because is the diameter. Also, because they both subtend arc . Therefore, by AA similarity, so we have or However, remember that area , so . Substituting this in gives us and then bash through algebra to get and we are done.
--Nosaj 19:39, 7 December 2014 (EST)
Formula for Circumradius
Where is the Circumradius, is the inradius, and , , and are the respective sides of the triangle. Note that this is similar to the previously mentioned formula; the reason being that .
Euler's Theorem for a Triangle
Let have circumradius and inradius . Let be the distance between the circumcenter and the incenter. Then we have