jmc

The surface area of the solid can be split into four pieces: the top and bottom, the curved side, and the flat side.

The top and bottom pieces are semicircles with radius 6; together, they add up to a circle with radius 6 and area $\pi (6^2)=36\pi$.

The curved side can be rolled out to a rectangle with height 10. The width of this rectangle is half the circumference of the base of the cylinder, which is $\frac{1}{2}\cdot 2\pi \cdot 6=6\pi$. Thus the area of the curved side is $10\cdot 6\pi =60\pi$.

The flat side is a rectangle with height 10. The width of this rectangle is the diameter of the cylinder, which is $6\cdot 2=12$. Thus the area of the flat side is $10\cdot 12=120$.

Finally, the total surface area of the solid is $36\pi +60\pi +120=\boxed{96\pi +120}$.