2005 Alabama ARML TST Problems/Problem 3

Problem

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

Let $R$ be the radius of the circumcircle and let $r$ be the radius of the incircle. Then we have $R^2-r^2=300$ . If the center of these two circles is $O$ , the vertices are $A, B$ and $C$ , and $M$ is the midpoint of side $AB$ , triangle $\triangle AMO$ is a $30^\circ-60^\circ-90^\circ$ right triangle, and its hypotenuse has length $R$ and its shorter leg has length $r$ . Thus $R = 2r$ . (There are many other arguments to get to this conclusion; for instance, $O$ is also the centroid of the triangle and $COM$ is a median, so $O$ trisects $CO$ and $R = CO = 2OM = 2r$ .)

Then $4r^2 - r^2 = 300$ so $r = 10$ and the side length of the triangle is equal to $10\sqrt 3$ .