For the triangle below, let x be the area of the circumcircle, and let y be the area of the incircle. Compute x - y.
In an equilateral triangle, the circumcenter and incenter coincide, and the radius of the circumcircle is equal to the length of the side of the triangle, while the radius of the incircle is equal to a/2√3.
The area of the circumcircle is given by πr^2, where r is the radius of the circumcircle, so x = πa^2/4.
The area of the incircle is given by πr^2, where r is the radius of the incircle, so y = πa^2/12√3.
Therefore, x - y = πa^2/4 - πa^2/12√3 = πa^2/4 - πa^2/4sqrt(3)
Hence, x - y = πa^2/4 - πa^2/4sqrt(3)
