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,
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