Module and Pitch: The One Number Two Gears Must Share

A gear has several diameters and only one of them matters for meshing. **The pitch circle** is where two meshing gears effectively roll on each other without slipping. It is the gear as far as kinematics is concerned — the ratio, the centre distance and the pitch line velocity are all about this circle. It is also invisible: there is nothing on the gear at that diameter. **The addendum circle** is the outside, over the tips. One module above the pitch circle for a standard tooth, so the outside diameter is m(z + 2). **The dedendum circle** is the root, at the bottom of the tooth spaces. 1.25 modules below the pitch circle, the extra quarter giving clearance for the tip of the mating tooth. **The base circle** is where the involute curve starts, and the next rungs need it. **Circular pitch** is the distance from one tooth to the next measured round the pitch circle: π times the module. It is the intuitive measure and it is awkward arithmetically, which is why module exists — module is circular pitch divided by π, chosen so the numbers come out round.

**Count the teeth.** Mark one with a pen and count round. It sounds trivial and it is the single most common error, because a gear with 41 teeth looks exactly like one with 40. **Measure the outside diameter** with a caliper. On an even tooth count, measure tip to tip straight across. On an **odd** count there is no tooth opposite a tooth, so a direct measurement reads low — measure over the bore or use the formula with the tooth count, and the arithmetic is more reliable than the caliper. **Then module = outside diameter divided by (teeth + 2)**, and round to the nearest standard. **Check the tooth height** as a confirmation: a standard full-depth tooth is 2.25 modules from root to tip. If the height does not agree with the module the diameter suggested, the gear is a stub tooth form or it has been cut with addendum modification. **Measure the face width and the bore** too, and write all of it down together. A gear identified but not recorded is a gear that will be identified again next month.

**Gears are cut with tools that are specific to the module**, and that is the whole reason for standardisation. A hob or a cutter cuts one module. A set of involute cutters for one module comes in eight numbers, each covering a range of tooth counts, because the correct involute shape changes with the number of teeth — a cutter for a 20-tooth gear is visibly a different shape from one for a 100-tooth gear of the same module. So a non-standard module means a special cutter, and a special cutter costs more than the machine time for the gear. **Prefer the standard series**, which is why the notebook's list looks like it does: 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10. The intermediate sizes exist and are much less available. **Inch gears are quoted in diametral pitch** — 24 DP, 20 DP, 16 DP — and are everywhere in older machinery and in model engineering. A 24 DP gear is module 1.058, which is not a standard module, so the two families genuinely do not interchange. **Check what the rest of the machine uses before choosing.** A gearbox with one odd module in it is a gearbox that cannot be repaired from stock.