= Gear lead
{title2=$L=\pi d/\tan|\beta|$}
<Gear lead> $L$ is the axial advance of one tooth helix over a complete turn around a cylindrical <helical gear>. It differs from <gear axial pitch> $p_x$, which steps between successive teeth at one circumferential position. With $N$ equally spaced tooth helices, $L=Np_x$. The lead magnitude does not specify whether the helix is right-handed or left-handed.<KHK Gear Nomenclature>[Technical reference §4.3, Figure4.7: one-turn lead and reference-cylinder helix]{ref}
Unwrapping a reference cylinder of diameter $d$ gives a right triangle with axial side $L$ and circumferential side $\pi d$. Since <gear helix angle> $\beta$ is measured from the axis,
$$
L=\frac{\pi d}{\tan|\beta|}.
$$
For $d=60\,\mathrm{mm}$ and $|\beta|=30\,{}^{\circ}$, $L=60\pi\sqrt3\,\mathrm{mm}\approx326.484\,\mathrm{mm}$. If $N=30$, then <gear axial pitch> is $10.8828\,\mathrm{mm}$; confusing the two distances would introduce a factor30.
A useful geometric check keeps the lead fixed while changing the chosen concentric cylinder. The helix's axial advance per revolution remains $L$, but its angle satisfies $\tan|\beta(r)|=2\pi r/L$. For the same lead on diameter $40\,\mathrm{mm}$, the angle is approximately $21.052\,{}^{\circ}$, rather than $30\,{}^{\circ}$. Thus a helix-angle specification needs its cylinder, normally the reference cylinder. Lead can remain finite when the physical tooth is much shorter than one complete turn: it describes the continued helix, not a requirement that the manufactured tooth cover a full revolution.
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