Derivation of Formula for Radius of Circumcircle

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derivation of formula
circumcenter
circumcircle
circumscribing circle

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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