Question 15
A block of wood has the shape of a right circular cylinder with radius $6$ and height $8,$ and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that arc $\overset{\frown}{AB}$ on that face measures $120^\circ.$ The block is then sliced in half along the plane that passes through point $A,$ point $B,$ and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of these unpainted faces is $a\cdot\pi + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a + b + c.$