Spur gears: module, diameters and center distance — formulas and calculator
Published October 8, 2026 — the formulas of a standard spur tooth from the module and the number of teeth, standard modules, center distance, minimum tooth count and a calculator for a gear pair.
In short: for a standard spur gear (20° pressure angle, no profile shift), everything follows from the module m and the number of teeth z: pitch diameter d = m × z, tip diameter da = m × (z + 2), root diameter df = m × (z − 2.5), center distance a = m × (z1 + z2) / 2. Two gears only mesh if they have the same module. Pick the module from ISO 54 series I: 1; 1.25; 1.5; 2; 2.5; 3; 4; 5; 6; 8; 10…
1. Spur gear formulas
Standard teeth per the ISO 53 basic rack (addendum m, dedendum 1.25 m), pressure angle α = 20°, no profile shift. All lengths in millimetres.
Example: m = 2, z1 = 20, z2 = 50. Pinion: d = 40, da = 44, df = 35 mm. Gear: d = 100, da = 104, df = 95 mm. Center distance 70 mm, ratio 2.5.
2. Calculator: a spur gear pair
3. Standard modules (ISO 54)
Prefer series I: cutters and off-the-shelf gears are easier to find there. In series II, the standard asks to avoid module 6.5. Conversion from inch gears: m = 25.4 / DP (diametral pitch).
4. Design rules to know
- Same module, same pressure angle: the condition for two gears to mesh.
- Minimum tooth count: about 17 at 20° without shift to avoid undercut (2 / sin² α ≈ 17.1); 14 in practice; fewer with a positive profile shift.
- Co-prime tooth counts: avoid an exact integer ratio (20 / 40) so each tooth meets every tooth of the other gear and spreads wear.
- Shaft connection: the gear hub is most often keyed; the section is chosen from the shaft diameter, see the DIN 6885 key chart, and the bore is toleranced with an ISO 286 fit.
- What this memo does not cover: strength calculation (bending and contact stress, ISO 6336), helical teeth and profile shift, which change diameters and center distance.
5. FAQ
What is the module of a gear?
It is the ratio of the pitch diameter to the number of teeth: m = d / z, in millimetres. It sets the tooth size: two gears can only mesh if they have the same module (and the same pressure angle).
How do you calculate the outside diameter of a pinion?
For a standard, unshifted tooth: da = m × (z + 2). A 20-tooth pinion at module 2 has a 40 mm pitch diameter and a 44 mm tip diameter.
How do you calculate the center distance of two gears?
a = m × (z1 + z2) / 2, for spur gears without profile shift. With m = 2, z1 = 20 and z2 = 50: a = 2 × 70 / 2 = 70 mm.
What is the minimum number of teeth on a pinion?
With a 20° pressure angle and standard teeth, about 17 teeth theoretically avoid undercut (2 / sin² 20° ≈ 17.1); in practice down to 14 teeth, or fewer with a positive profile shift.
How do you convert diametral pitch to module?
m = 25.4 / DP. A DP 10 gear corresponds to a 2.54 mm module, which is not a standard module: inch and module gears are generally not interchangeable.