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Gear axial pitch $p_x$ is the axial repeat distance between corresponding flanks of successive teeth of a cylindrical helical gear. The measurement follows a line parallel to the rotation axis on a concentric cylinder. It is not the axial advance of one tooth helix through an entire revolution; that larger distance is gear lead $L$. For $N$ equally spaced teeth, $L=N{p}_x$.^[ref]

On the developed cylinder, neighboring tooth traces separated transversely by gear transverse pitch $p_t$ intersect a fixed circumferential coordinate at axial positions separated by $p_x$. Their slope givesFor $d=60\mathrm{mm}$, $N=30$ and gear helix angle $|\beta |=30{}^{\circ }$, $p_t=2\pi \mathrm{mm}$ and $p_x=2\pi \sqrt{3}\mathrm{mm}\approx 10.8828\mathrm{mm}$. The corresponding gear normal pitch is only $5.4414\mathrm{mm}$. These two distances differ by a factor of two in this example because $\sin 30{}^{\circ }=1/2$.

For an engaged gear face width $b=20\mathrm{mm}$, the ideal gear overlap ratio is $b/p_x\approx 1.8378$: successive tooth-contact traces span about1.84 axial pitches across the face. Holding $p_t$ fixed while letting $\beta$ tend to zero makes $p_x$ diverge. Straight spur gear tooth traces do not cross an axial generator successively, so a finite axial-pitch interpretation does not survive that limit. Helix hand determines orientation; the positive pitch magnitude alone does not specify hand.

The gear helix angle $\beta$ measures a cylindrical gear tooth's inclination to the rotation axis, normally on the reference cylinder. A straight spur gear has $\beta =0$; a helical gear has a nonzero inclination and a specified hand. The cylinder and reference direction matter because another measurement plane or cylinder radius can give another angle.^[ref]

Let $L$ be the positive gear lead, the axial advance of one tooth helix in a complete turn, and let the reference diameter be $d$. Unwrapping that cylinder produces a right triangle with circumferential side $\pi d$ and axial side $L$:For $d=60\mathrm{mm}$ and $|\beta |=30{}^{\circ }$, $L\approx 326.484\mathrm{mm}$. Doubling the lead on that same cylinder gives angle approximately $16.102{}^{\circ }$, rather than15degrees: the relation involves a tangent, not a linear angle scaling.

With $N$ equally spaced teeth, gear axial pitch is $p_x=L/N=p_t/\tan |\beta |$, where $p_t$ is gear transverse pitch. For nonzero $\beta$, this also gives $p_x=\pi m_n/\sin |\beta |$. The corresponding axial-pitch quantity diverges in the straight-tooth limit; it is not a finite replacement for the transverse tooth spacing of a spur gear.

Larger helix angle changes reference diameter at fixed gear normal module and tooth count, increases axial thrust for a fixed tangential load, and increases face-overlap geometry at fixed face width. These consequences must be evaluated together with gear contact ratio and support loads. A magnitude such as30degrees does not by itself specify right versus left hand, nor the thrust direction under a particular applied torque.

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.