AMC 12 2006a Problem 22

A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$?

$\mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3} \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2} \mathrm{(C) \ } 2\sqrt{6}+\sqrt{3} \mathrm{(D) \ } 3\sqrt{2}+\sqrt{6} \mathrm{(E) \ } 6\sqrt{2}-\sqrt{3}$