Actually

is the volume of a cylinder.

is the Surface Area.

If you think about it you'll see why. You need the area of the circle on top, and bottom which is * 2.

Now the circumference of those circles is and we need to add all of the circumferences from top to bottom. So we multiply by h. This gives us the edge all the way around connecting the top to the bottom.Now this gives the total Surface Area.

Thinking about it I forgot to add. If you cut a hole in the cylinder you have to consider adding up the circumferences of the inside you cut out since they are not on the surface.

I think your drawing is like a mason jar lid. A cylinder with a hole all the way through from top to bottom. So what pieces from the total surface area are missing? Figure that out and subtract it from the above formula.

N.B: Sorry forgot to add. Since we are cutting a hole through the middle of this thing. We need to find the circumferences of the inside ring and multiply by h. Then ADD that to the total, since that is now part of the surface and wasn't before.