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2005 Alabama ARML TST Problems/Problem 3

Problem

The difference between the areas of the circumcircle and incircle of an equilateral triangle is $\displaystyle 300\pi$ square units. Find the number of units in the length of a side of the triangle.

Solution

Let $R$ be the radius of the circumcircle, and let $r$ be the radius of the incircle. Then we have $R^2-r^2=300$. But by similarity in the equilateral triangle $r=2r$, thus $r=10$; side length is equal to $10\sqrt{3}$.

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