The calculations performed by the rolling wire-strip calculator were derived by examining experimental data. We might not have considered all of the necessary variables. Anyway, here’s how it currently works.
Let’s suppose we start with square wire, with side $S$, and we roll it to thickness $t$. Then we find that the wire’s width is \[ w = \sqrt{\frac{S^3}{t}} \,\text{.} \] Rearranging, we find that \[ S = \sqrt[3]{w^2 t} \,\text{.} \] For round wire, we assume that the cross-section area is the important bit, so a round wire with diameter $D$ ought to work as well as square wire with side $S$ if $S^2 = \pi D^2/4$, i.e., \[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \,\text{.} \] Volume is conserved, so if the original and final wire lengths are $L$ and $\ell$ respectively, then \[ L S^2 = \ell w t \,\text{,} \] and hence \[ L = \frac{\ell w t}{S^2} \,\text{.} \] Finally, determining the required initial stock length $L_0$ given its side $S_0$ (for square stock) or diameter $D_0$ (for round) again makes use of conservation of volume: \[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \,\text{.} \]
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