Gear Pitch Diameter

A gear's pitch diameter is the diameter of a cylinder that is not there. Imagine the two gears with their teeth removed and replaced by smooth rollers pressed together, turning without slipping: the diameters of those imaginary rollers are the pitch diameters, and everything else about the pair — the ratio, the centre distance, the tooth load — is measured against them. The teeth exist only to stop the rollers from slipping.

So the pitch circle is not any surface you can put a caliper on. It sits partway down the tooth, between the tip and the root. Measuring across the tips gives the OUTSIDE diameter, which on a standard full-depth tooth is larger by two addenda, or $d_o = d + 2m$. Reading that measurement as the pitch diameter is the most common error in gear work after the one below, and it quietly inflates the pitch diameter of a small pinion by a real percentage — a 20-tooth, module-2 gear measures 44 mm over the tips and has a pitch diameter of 40.

Two conventions, and they are reciprocals

The metric world defines the module, $m = d/N$, in millimetres. It is a length: it is roughly the size of one tooth, and a bigger module means bigger teeth. Multiply it by the tooth count and you have the pitch diameter, which is the relation this page computes, and it is dimensionally honest in any unit you like.

North America defines diametral pitch, $P = N/d$, and here is the trap: that equation is only true with $d$ in inches. Diametral pitch is teeth per inch of pitch diameter, so the inch is baked into the number the way it is baked into "pounds per square inch". It is a count per unit length, which means it runs backwards from the module — a bigger diametral pitch means SMALLER teeth. A 4 DP gear is coarse; a 32 DP gear is fine enough for a clock.

The bridge between them is exact, because an inch is exactly 25.4 mm:

\[ m \,[\text{mm}] = \frac{25.4}{P} \qquad P = \frac{25.4}{m\,[\text{mm}]} \]

So a 4 diametral pitch gear is module 6.35 mm, and module 2 is 12.7 DP. Notice that the standard series do not line up: standard modules are 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 and 10 mm, standard diametral pitches are whole numbers, and almost none of them coincide. That is why a metric gear and an inch gear of nominally similar size will not mesh, and why the two systems remain stubbornly separate on the shop floor.

The number 17

Standard 20° full-depth teeth undercut below about 17 teeth on the pinion. The cutter, sweeping through, removes material from the flank near the root — exactly the section carrying the bending load — and the tooth gets weaker and the contact rougher. It can be done deliberately with profile shift, a higher pressure angle, or simply by accepting the weaker tooth in a lightly loaded drive, but 17 is the number to have in your head when someone proposes a 12-tooth pinion. AGMA's public geometry documents and NASA's gearing design summaries both treat it as the practical floor for standard teeth.