I don't know if you've solved your problem yet, but I just tried to cut a sphere out of a screen. There are a bunch of ways to do it - but the simplest is the one I think you're describing, where you just cut longitudinal sections of the sphere. A buckyball will often give better results for something stretchy like fabric. But anyway, here's the idea for the sections.
Think of each section as a constant angle slice of the surface of the sphere. So to draw a section, find the distance subtended on the surface of the sphere at that given latitude for the given number of sections. For example,
D = 9
C = 28.274
8 arcs. Each arc is 45 degrees (probably a little big), and so covers 45/360ths = 1/8th of the circumference, so it is 3.534 at the middle, and 0 at the ends. Now to find the width of the section at other locations, find the circumference of the circular slice of the sphere at that height. As you go from 0 to 90 degrees from the pole to the equator, the diameter of one slice of the sphere is d = D * sin(theta). Pi*d is the circumference at that latitude. 1/8th of that is the width of the slice at that angle.