= Gear circular pitch
{title2=$p=\pi m$}
= Gear transverse pitch
{synonym}
<Gear circular pitch> is the arc distance between corresponding points on successive <gear> teeth, measured on the <gear reference circle>. If that circle has diameter $d$ and tooth count $N$, its circumference contains $N$ repetitions, giving
$$
p=\frac{\pi d}{N}=\pi m,
$$
where $m$ is <gear module>. The length is measured along the reference arc, rather than as a straight chord between tooth positions.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §2, printed pp.12–14, reference pitch and $p=\pi m$]{ref}
For a <spur gear> with $m=2\,\mathrm{mm}$ and $N=30$, reference diameter is $60\,\mathrm{mm}$, <gear circular pitch> is $2\pi\,\mathrm{mm}\approx6.283\,\mathrm{mm}$, and angular tooth spacing is $2\pi/30$. The corresponding chord on the reference circle is
$$
\ell=2(30\,\mathrm{mm})\sin(\pi/30)\approx6.272\,\mathrm{mm}.
$$
The small numerical difference matters when a dimensional inspection is intended to measure reference arc spacing. Increasing the tooth count while holding <gear module> fixed enlarges the <gear reference circle> but preserves <gear circular pitch>.
For an <involute> tooth profile, the <gear base circle> provides another spacing, the <gear base pitch>. Since $d_b=d\cos\alpha$, with reference <gear pressure angle> $\alpha$,
$$
p_b=\frac{\pi d_b}{N}=p\cos\alpha.
$$
At $\alpha=20\,{}^{\circ}$, the preceding example gives $p_b\approx5.904\,\mathrm{mm}$. Compatible contact must preserve the spacing of successive tooth flanks along their <gear line of action>; this explains why equal <gear circular pitch> with incompatible <gear pressure angles> is not enough to specify a mating pair. Tooth thickness and <backlash> are additional quantities: <gear circular pitch> includes an entire tooth-and-space repetition.
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