University of South Carolina High School Math Contest/1993 Exam/Problem 29
Contents
[hide]Problem
If the sides of a triangle have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle?

Solutions
Solution 1
One simple solution is using area formulas: by Heron's formula, a triangle with sides of length 2, 3 and 4 has area . But it also has area
(where
is the circumradius) so
.
Solution 2
Alternatively, let vertex be opposide the side of length 2. Then by the Law of Cosines,
so
. Thus
. Then by the extended Law of Sines,
.