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 $\overarc{AB}$ 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 $a+b+c$.

--Credit to Royalreter1 and chezbgone2 For The Diagram--
[2015 AIME I 15.png](https://artofproblemsolving.com/wiki/index.php/File:2015_AIME_I_15.png)
Label the points where the plane intersects the top face of the cylinder as $C$ and $D$, and the center of the cylinder as $O$, such that $C,O,$ and $A$ are collinear. Let $N$ be the center of the bottom face, and $M$ the midpoint of $\overline{AB}$. Then $ON=4$, $MN=3$ (because of the 120 degree angle), and so $OM=5$. 
Project $C$ and $D$ onto the bottom face to get $X$ and $Y$, respectively. Then the section $ABCD$ (whose area we need to find), is a stretching of the section $ABXY$ on the bottom face. The ratio of stretching is $\frac{OM}{MN}=\frac{5}{3}$, and we do not square this value when finding the area because it is only stretching in one direction. Using 30-60-90 triangles and circular sectors, we find that the area of the section $ABXY$ is $18\sqrt{3}\ + 12 \pi$. Thus, the area of section $ABCD$ is $20\pi + 30\sqrt{3}$, and so our answer is $20+30+3=\boxed{053}$.