Derivation of Formula for Radius of Circumcircle

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

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

where At is the area of the inscribed triangle.
 

Derivation:
Figure for derivation of radius of circumcircleIf you have some questions about the angle θ shown in the figure above, see the relationship between inscribed and central angles.
 

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

$\sin \theta = \dfrac{a}{2R}$
 

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

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

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

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

 

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