Wildgoose, I don't think we're talking about the same thing. Here's a diagram. In order to get the red square to appear square on the cylinder, it will need to be stretched to the size of the blue rectangle.
Go-C, after doing the math, 10% is pretty close to what the math comes back with. It'll vary some depending on size of cup/cylinder and size of image but 10% is a good ballpark.
After some more thinking I think I decided I want to shrink the height rather than change to width so here is my new equation.
H = h/w*D*sin(w/D)
h - height of image
w - width of image
D - diameter of cup
H - new height for image