Derivation of Formula for Radius of Circumcircle

The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by
 

where A_t is the area of the inscribed triangle.
 

Derivation:
If you have some questions about the angle θ shown in the figure above, see the relationship between inscribed and central angles.
 

From triangle BDO
$\sin \theta = \dfrac{a/2}{R}$

$\sin \theta = \dfrac{a}{2R}$
 

A_t = area of triangle ABC
$A_t = \frac{1}{2}bc \sin \theta$

$A_t = \frac{1}{2}bc \left( \dfrac{a}{2R} \right)$

$A_t = \dfrac{abc}{4R}$