= Face gear
A \b[face gear] is a <gear> whose working teeth occupy an annular face and mesh with a mating cylindrical pinion. A common arrangement turns the drive through approximately a right angle; other shaft angles and tapered configurations are possible. Its working surface geometry must maintain compatible tooth contact, rather than simply placing ordinary cylindrical teeth on a flat disk.
Tooth counting still supplies an ideal speed relation. If the pinion has $N_p$ teeth and the <face gear> $N_f$, equality of tooth-passage rates gives
$$
N_p|\omega_p|=N_f|\omega_f|.
$$
An original 20-tooth pinion driving a 100-tooth <face gear> at $1500$ revolutions per minute therefore gives $300$ revolutions per minute at the output. Rotation signs require a stated orientation of the two different shaft axes; the scalar minus sign for an external parallel-axis <spur gear> pair cannot be copied without that convention.
The annular layout also explains a geometric design issue. With $N$ tooth positions, angular spacing is $2\pi/N$, while arc spacing on a ring of radius $r$ is
$$
p(r)=\frac{2\pi r}{N}.
$$
For $N=80$, radii $40$ and $60\,\mathrm{mm}$ give arc spacings about $3.142$ and $4.712\,\mathrm{mm}$. This radial change does not prescribe two interchangeable <gear modules>; it shows why a generated face-tooth surface cannot be understood as a single extruded cylindrical tooth profile.
A primary 2002 program built a scaled-power <split-torque transmission> with two opposed <face gears>, two input pinions and two idlers.<Development of Face Gear Technology for Industrial and Aerospace Power Transmission by Gregory F. Heath and others>[Introduction, printed pp. 1–2, proof-of-concept arrangement; Conclusions, printed p. 84, off-90-degree tapered configuration]{ref} Multiple meshes make the geometry useful for distributing and recombining load, but also make alignment and compatibility important.
A bench rotation check can test the tooth-count ratio. A <gear contact marking test>, dimensional inspection and load-sensitive measurements address different questions about the manufactured surfaces and their operation. Correct output speed alone does not establish an acceptable <gear contact pattern>, tooth <stress> or <gear efficiency>.
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