
This is the way I calculate the length of wire required for a given ring. The formulas shown will calculate the length of the wire connecting the two sides along a fractional segment of the circumference, calculated as a function of the diameter, Pi, the number of leads, and the number of bights. Note in the second formula that the quantity labeled "seg" (for the sake of brevity) is the same, both being length of the bottom of the triangle superimposed on the ring.
With a very simple ring, open of weave and narrow with respect to its size, the second forumula is the more accurate, but as the rings grow larger and have a greater proportion of the wire following a straight path, the first forumula will give superior results. Since the formulas only calculate the length of one such path, and the total length of wire is a multiple of that path, the errors accumulate.
Naturally, both of the formulas are assuming a perfectly smooth wire of zero diameter, following the exact path without deviation  so I always cut the wire a bit longer than I think I'll need, to give me something to hang onto with the pliers.
I have written up a somewhat more extensive discussion of this elsewhere, if you are interested.



