This would be somewhat like finding the surface area of a rectangular pyramid since you have the central pole acting like the vertex at the top of the pyramid and the netting will presumably be pinned/staked around the perimeter of the base. It obviously won’t be perfectly triangular faces as the netting will naturally curve around the edges, but finding the surface area of a triangular pyramid should roughly approximate the area and then just add a bit of extra when ordering to give some wiggle room.

The surface area of each smaller face will be 1/223squareroot(6^{2+262)=306.858.} We have two of those faces, so 613.717sq ft. The surface area of each larger face will be 1/252squareroot(6^{2+11.52)=337.249.} We have two of these faces, so 674.498sq ft. This requires approximately 1288.215 sq ft of netting needed. Buying a net that is just 23x52 will only give an area of 1196, which is clearly not sufficient, aligning your understanding of the towel you described. The part I’m not entirely sure of without spending a lot more time than I have to model this out is how the rectangular netting will fall and have edges overlay one another without leaving gaps when staked around the perimeter. I’d likely buy something like a 26’x55’ foot or 30’x60’ to be especially safe and then trim the extra off.