2005 Alabama ARML TST Problems/Problem 3
Problem
The difference between the areas of the circumcircle and incircle of an equilateral triangle is square units. Find the number of units in the length of a side of the triangle.
Solution
Let be the radius of the circumcircle, and let
be the radius of the incircle. Then we have
. But by similarity in the equilateral triangle
, thus
; side length is equal to
.