I thought some more and I am happy with the equation as is (with your correction).
If instead of normal spiral half-lap wrap, you examine an idealized version it is simpler to see. The idealized version would be short strips of length CP wrapped around the cirfumference to form a loop. Then next loop overlaps the previous loop by a half tape width. It is easy to see the expression would apply to this ideal geometry. And you have two thicknesses of tape on the entire surface of the coil (neglecting gaps).
Now instead look at a real half lap with spiral, you still have two thickness of tape at all points on the coil, so you still have the same total volume of tape, so you must have the same length of tape and the equation above is correct.
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