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.
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,
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