= Gear module
{title2=$m=d/N$}
The <gear module> specifies the size of metric <gear> teeth through the <gear reference circle>. For a <spur gear> with reference diameter $d$ and $N$ teeth,
$$
m=\frac dN,\qquad d=mN.
$$
The tooth count is dimensionless, so <gear module> has units of length, conventionally millimetres. A larger <gear module> means a larger reference diameter at fixed tooth count. It does not mean a larger <gear ratio>, which depends on the tooth counts of the meshing pair.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §2, printed pp. 12–15, module and equation (2.8)]{ref}
The <gear circular pitch> $p$ is the arc length between equivalent points on adjacent teeth, measured along the <gear reference circle>. Dividing the circumference by the tooth count gives
$$
p=\frac{\pi d}{N}=\pi m.
$$
For example, a <spur gear> with $m=1.5\,\mathrm{mm}$ and $N=24$ has reference diameter $36\,\mathrm{mm}$ and <gear circular pitch> $1.5\pi\,\mathrm{mm}\approx4.712\,\mathrm{mm}$. A 48-tooth <gear> of the same <gear module> has diameter $72\,\mathrm{mm}$ but the same tooth spacing. Their standard axis separation is $(36+72)/2=54\,\mathrm{mm}$, and their reduction <gear ratio> is $48/24=2$.
Conversely, two <gears> can have the same diameter without compatible tooth spacing: $d=36\,\mathrm{mm}$ can describe 24 teeth of <gear module> $1.5\,\mathrm{mm}$ or 18 teeth of <gear module> $2\,\mathrm{mm}$. Such teeth do not form the specified standard pair. For ordinary compatible <spur gears>, matching <gear module> is necessary but does not by itself specify the complete tooth geometry: the <gear pressure angle> and other tooth details must also agree. The definition uses reference diameter, not the diameter through the tooth tips or the <operating pitch circle> of a particular mounting.
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