Circumradius
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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.
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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,
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