I haven't been reading these forums for long, but this is the strangest that I have seen. I'm not sure that anyone here is really interested in how the area of contact is calculated but here is my take. I've been a student of pool all of my life and a decent but certainly not great player ( I was a 7 in Busch league pool and was collegiate champion at Tulane University for three years in the 60"s and 70's). I practiced engineering for 40 years and was in college for 7 prior to that getting BS and Masters in mechanical engineering/thermodynamics so I have some mathematical cred, although I've probably forgotten more of what I learned than what I remember. So, unless one is trying to obtain the answer to 5 (maybe more) significant digits/figures (

Significant figures - Wikipedia ) the contact area due to the compression of the ball and slate are negligible. If we remove that from consideration, the contact area would be the area of the cap of the sphere (ball) calculated with r as the radius of the ball and h as the depth of the depression/compression. See this Wiki site for equation

Spherical cap - Wikipedia Of course, to actually make that calculation, one would need to calculate the compression of the cloth based on the physical characteristics of the cloth and the weight and radius of the ball. I expect we could determine how that calculation could be made, but I doubt that getting the characteristics of the cloth for any given situation to make that calculation would be possible. One could probably determine the compression (h in the calculation) of the cloth by the ball by measuring it if one had the proper instruments.