AMC10 2003 10A, Problem 17

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of  its circumscribed circle. What is the radius, in inches, of the circle?

Solution: Let $s$ be the length of a side of the equilateral triangle and let $r$ be the radius of the circle. In a circle with a radius $r$, the side of an inscribed equilateral triangle is $r\sqrt{3}$. So $s=r\sqrt{3}$. The perimeter of the triangle is $3s=3r\sqrt{3}$ The area of the circle is $\pi r^{2}$ So: $\pi r^{2} = 3r\sqrt{3}$ $\pi r=3\sqrt{3}$ $r=\frac{3\sqrt{3}}{\pi} \Rightarrow\fbox{\frac{3\sqrt{3}}{\pi}}$

Answer: \frac{3\sqrt{3}}{\pi}