Circumradius
This article is a stub. Help us out by expanding it.
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.
Contents
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 . Also,
Proof
Proof: [asy] pair O, A, B, C, D; O=(0,0); A=(-5,1); B=(1,5); C=(5,1); dot(O); dot (A); dot (B); dot (C); draw(circle(O, sqrt(26))); draw(A--B--C--cycle); D=-B; dot (D); draw(B--D--A); label("", A, W); label("", B, N); label("", C, E); label("", D, S); label("", O, W); pair E; E=foot(B,A,C); draw(B--E); dot(E); label("", E, S); draw(rightanglemark(B,A,D,20)); draw(rightanglemark(B,E,C,20)); [/asy]
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.
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