 Gear lead

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, $L=N{p}_x$. The lead magnitude does not specify whether the helix is right-handed or left-handed.^[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,For $d=60\mathrm{mm}$ and $|\beta |=30{}^{\circ }$, $L=60\pi \sqrt{3}\mathrm{mm}\approx 326.484\mathrm{mm}$. 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.