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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.

Gear contact ratio measures how much successive tooth engagements overlap. For an external spur gear pair, its transverse value iswhere $L$ is usable gear path of contact length along the gear line of action and $p_b$ is gear base pitch. One tooth pair remains engaged while contact advances through length $L$; successive pairs begin after an advance of $p_b$. Thus ${ϵ}_{\alpha }$ is also the ideal mean number of tooth pairs simultaneously engaged.^[ref]

For a tip-limited, noninterfering external involute pair with tip radii $r_{a1},r_{a2}$, base radii $r_{b1},r_{b2}$, axis separation $a$ and operating pressure angle ${\alpha }_w$, right-triangle geometry givesFor an original example, use module $2\mathrm{mm}$, tooth counts 20 and 40, reference angle $20{}^{\circ }$ and standard separation $60\mathrm{mm}$. With tip radii $22$ and $42\mathrm{mm}$, the formula gives $L\approx 9.6546\mathrm{mm}$ and $p_b\approx 5.9043\mathrm{mm}$, hence ${ϵ}_{\alpha }\approx 1.6352$.

In the ideal periodic timing model, two tooth pairs are engaged for about $63.52\%$ of a tooth-passage period and one pair for about $36.48\%$. This does not mean that force is divided equally between engaged pairs: tooth flexibility and contact geometry affect load sharing. A value below one would leave gaps in ideal engagement, while a value exactly one leaves no overlap margin for tolerances and deformation.

The plot shows contact timing, rather than physical tooth shapes or force sharing. For helical gears, the transverse measure must additionally be distinguished from gear overlap ratio caused by the inclination of teeth across the gear face. Finally, real usable contact can be shortened by modified or unavailable flank portions; the tip-circle formula is conditional on its stated geometric assumptions.

Gear effective face width is the usable axial width participating in a stated cylindrical gear mesh. It must be distinguished from each component's nominal gear face width, and from a claim that every part of that width carries equal load. The normal/transverse tooth geometry and gear overlap ratio use this engaged width, while gear face load distribution describes how the load actually occupies it.^[ref]

An elementary geometric model treats the two toothed widths as intervals $I_1=[a_1,b_1]$ and $I_2=[a_2,b_2]$ on their common axial coordinate. Their available overlap isFor one20mm-wide gear occupying $[-10,10]\mathrm{mm}$ and a16mm-wide mate centered at $6\mathrm{mm}$, the second interval is $[-2,14]\mathrm{mm}$. Their overlap is only12mm. The mean line load for a hypothetical uniformly shared normal force of1200N is then100N/mm, rather than the60N/mm obtained by using the first gear's entire nominal width.

With gear normal module $m_n=2\mathrm{mm}$ and gear helix angle $|\beta |=30{}^{\circ }$, gear axial pitch is $p_x=4\pi \mathrm{mm}$. The12mm overlap gives geometric gear overlap ratio $b_{\mathrm{eff}}/p_x=3/\pi \approx 0.9549$, compared with $5/\pi \approx 1.5915$ for20mm. Axial positioning can therefore alter overlap without changing the ideal gear ratio.

The interval calculation assumes the flanks remain usable throughout their common extent. Relief, damaged regions or partial contact can require a more specific working-width convention. Uneven loading is not generally corrected by shrinking width until a uniform-load formula looks plausible; a stated load-distribution model is needed to predict local loading.

Gear face width $b$ describes the width of the toothed portion. For a cylindrical spur gear or helical gear, it is measured parallel to the rotation axis, rather than along the slanting tooth trace. A projecting hub need not be part of the toothed width. Manufacturer drawings distinguish face width, hub width and total length as separate dimensions.^[ref]

The available width of one gear is not necessarily the gear effective face width of a pair. For two aligned cylindrical gears with equal width $b$ and relative axial offset $e$, a simple interval-overlap model givesAn original $b=20\mathrm{mm}$ pair offset by $4\mathrm{mm}$ has only $16\mathrm{mm}$ of common axial extent. If a hypothetical normal load $1000\mathrm{N}$ were uniformly spread across that extent, mean load per axial width would rise from $50$ to $62.5N/mm$. This is bookkeeping for an ideal uniform load, not a tooth-strength prediction. Actual gear face load distribution also depends on alignment and deformation.

For helical gears, width also enters gear overlap ratio:With gear normal module $m_n=2\mathrm{mm}$ and gear helix angle $|\beta |=30{}^{\circ }$, reducing engaged width from $20$ to $16\mathrm{mm}$ reduces ideal overlap from $1.5915$ to $1.2732$. The overall gear ratio is unchanged. Width therefore affects engagement and loading even when tooth counts and shaft speeds remain the same.^[ref]

Gear overlap ratio measures the additional engagement overlap caused by a helical gear's tooth inclination across its gear face width. It is distinct from the transverse gear contact ratio, which measures contact length within one transverse section. For face width $b$, reference gear helix angle $\beta$ and gear normal module $m_n$,where $m_t$ is gear transverse module and $p_x$ is gear axial pitch.^[ref]

Unwrapping the reference cylinder explains the ratio: moving across axial width $b$ shifts the tooth pattern circumferentially by $b\tan |\beta |$. Dividing this shift by gear transverse pitch $\pi m_t$ counts how many tooth spacings the engagement progresses across the face.

For an original example, $b=20\mathrm{mm}$, $m_n=2\mathrm{mm}$ and $|\beta |=30{}^{\circ }$ giveDoubling the face width doubles this geometric overlap if the other parameters stay fixed. At the spur gear limit $\beta =0$, the overlap contribution tends to zero.

In the ideal cylindrical-helical model, total geometric contact ratio is the sum of transverse and axial contributions, ${ϵ}_{\gamma }={ϵ}_{\alpha }+{ϵ}_{\beta }$. This does not specify a uniform division of load: flexibility, alignment and the usable flank region affect actual contact. Increasing helix angle can improve overlap while also increasing gear axial force; increasing width also changes the alignment and support demands. These are related geometric tradeoffs, not a guarantee of a specified noise level or load capacity.