Spur gear calculation: module, ratio and the numbers that make gears mesh.
Two gears only mesh if their teeth are the same size. That single fact — captured by the module — sits behind every spur gear calculation. Here is how module, pitch diameter, gear ratio, and centre distance connect, with a worked example you can reuse.
Module: the one number that decides everything
In metric gear design, module (m) defines tooth size. It is the pitch diameter divided by the number of teeth, in millimetres:
where d = pitch diameter (mm), z = number of teeth.
The golden rule: two gears must have the same module to mesh. A module-2 pinion will not run with a module-2.5 gear, full stop. Modules are standardised — ISO 54 lists preferred values of 1, 1.25, 1.5, 2, 2.5, 3, 4, 5 and 6 mm. Always design to a standard module so you can buy stock cutters and off-the-shelf gears.
If you work in imperial units you will meet diametral pitch (DP) instead, measured in teeth per inch. The two are inversely related: module = 25.4 / DP.
Pitch diameter
The pitch circle is the imaginary circle where two gears effectively roll on each other without slipping. Its diameter is:
Pitch diameter is the working diameter for almost every other calculation — it sets centre distance, pitch line velocity, and the bending load on the tooth. Note it is not the outside diameter you measure with a caliper; the tips of the teeth sit one module above the pitch circle.
Gear ratio
The gear ratio links input and output speed and torque. For a driver (pinion) with z1 teeth driving a gear with z2 teeth:
A ratio of 3 means the output turns at one-third the speed and roughly three times the torque (before friction losses). For the speed and torque side in depth, see our companion guide on gear ratio calculation. For a single spur pair, keep the ratio at or below about 6–8; beyond that, use a two-stage train so the gears stay a sensible size.
Centre distance
For two meshing gears of the same module, the centre distance is simply the sum of the pitch radii:
This is the number your gearbox housing has to hold to within a few hundredths of a millimetre. Get it wrong and you either jam the teeth (too close) or get backlash and noise (too far). Profile-shifted gears change this slightly, but for standard gears the formula above is exact.
Tooth dimensions (standard 20° full-depth)
| Feature | Formula |
|---|---|
| Addendum (ad) | 1 × m |
| Dedendum (bd) | 1.25 × m |
| Whole depth | 2.25 × m |
| Outside diameter (da) | d + 2m = m(z + 2) |
| Root diameter (df) | d − 2.5m = m(z − 2.5) |
| Circular pitch (p) | π × m |
| Base circle (db) | d × cos(20°) |
To avoid undercutting on a 20° pressure-angle gear, keep the pinion tooth count at 17 or more. Below that you need profile shift to stop the cutter gouging the tooth root.
A worked example: conveyor drive
A packaging line needs a 3:1 reduction. The motor pinion has z1 = 20 teeth, and we choose module m = 2.5 mm.
- Driven gear teeth: z2 = i × z1 = 3 × 20 = 60 teeth.
- Pinion pitch diameter: d1 = 2.5 × 20 = 50 mm.
- Gear pitch diameter: d2 = 2.5 × 60 = 150 mm.
- Centre distance: a = 2.5 × (20 + 60) / 2 = 100 mm.
- Pinion outside diameter: 2.5 × (20 + 2) = 55 mm.
- Gear outside diameter: 2.5 × (60 + 2) = 155 mm.
The pinion has 20 teeth, comfortably above the 17-tooth undercut limit, so no profile shift is needed. The housing bore centres must be held at 100 mm ±0.03 mm to keep backlash within spec.
Common mistakes
- Mixing module and DP. A metric module-2 and an imperial 12-DP gear look similar but will not mesh.
- Measuring outside diameter as pitch diameter. The OD is two modules larger than the pitch diameter; using it throws off centre distance.
- Fewer than 17 pinion teeth without profile shift. Causes undercutting, weakening the tooth root.
- Forgetting backlash. Real gearboxes need a small backlash allowance; cutting to exact centre distance with zero clearance leads to binding and heat.
For the strength side of gear design, pair this with a fastener and torque reference for the gearbox assembly, and a material weight estimate when costing the blanks.
Frequently asked questions
What is module in a spur gear?
Module m = d / z is the ratio of pitch diameter to tooth count, in mm. It defines tooth size, and meshing gears must share the same module.
How do you calculate gear ratio?
Gear ratio i = z2 / z1, the driven gear teeth divided by the driver teeth. It equals the speed reduction and the torque multiplication.
How do you calculate centre distance?
For equal-module gears, a = m × (z1 + z2) / 2 — the sum of the two pitch radii.
How many teeth avoid undercutting?
For a standard 20° pressure angle, keep the pinion at 17 teeth or more, or apply profile shift below that.