# University of South Carolina High School Math Contest/1993 Exam/Problem 29

## Contents

## 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, .