Enter the module, the number of teeth and the angles, and the calculator gives you the pitch, tip and root diameters, centre distance, gear ratio, contact ratio and the span measurement for checking gears in the shop. It works for spur and helical gears. Only have an old gear? You can work out the module from the sample too.
| mm | Pinion · z = 18 | Gear · z = 45 |
|---|---|---|
| Pitch diameter d | 36.000 | 90.000 |
| Tip diameter da | 40.000 | 94.000 |
| Root diameter df | 31.000 | 85.000 |
| Base diameter db | 33.829 | 84.572 |
| Span measurement W | 15.265 (over 3 teeth) | 27.830 (over 5 teeth) |
The tooth count is above the theoretical undercut limit (17.10): the gears can be cut without profile shift.
Have a spur gear and no drawing? Measure the tip diameter with calipers and count the teeth.
m = da / (z + 2). For worn or profile-shifted gears, confirm the module with a span measurement.
These are the formulas the calculator uses, for the ISO 53 basic rack and gears with no profile shift. For helical gears, m is the normal module and the diameters use the transverse module mt = m / cos β.
| Quantity | Symbol | Formula |
|---|---|---|
| Module | m | m = p / π = d / z |
| Pitch diameter | d | d = m · z (helical: d = m · z / cos β) |
| Tip (outside) diameter | da | da = d + 2 · m |
| Root diameter | df | df = d − 2.5 · m |
| Base diameter | db | db = d · cos α |
| Circular pitch | p | p = π · m |
| Whole depth | h | h = 2.25 · m |
| Tooth thickness on the pitch circle | s | s = π · m / 2 |
| Centre distance | a | a = (d1 + d2) / 2 = m · (z1 + z2) / 2 |
| Gear ratio | i | i = z2 / z1 = n1 / n2 |
| Span over k teeth | W | W = m · cos α · [π · (k − 0.5) + z · inv α] |
| Diametral pitch (imperial) | DP | DP = 25.4 / m |
Gear modules are standardised so that tools and gears fit together. ISO 54 series I is preferred; below 1 mm are the common fine modules from DIN 780. The chart gives the circular pitch, the whole depth and the diametral pitch equivalent.
| Module m (mm) | Circular pitch π·m (mm) | Whole depth 2.25·m (mm) | Diametral pitch 25.4/m |
|---|---|---|---|
| 0.5 | 1.571 | 1.125 | 50.80 |
| 0.6 | 1.885 | 1.350 | 42.33 |
| 0.7 | 2.199 | 1.575 | 36.29 |
| 0.8 | 2.513 | 1.800 | 31.75 |
| 0.9 | 2.827 | 2.025 | 28.22 |
| 1 | 3.142 | 2.250 | 25.40 |
| 1.25 | 3.927 | 2.813 | 20.32 |
| 1.5 | 4.712 | 3.375 | 16.93 |
| 2 | 6.283 | 4.500 | 12.70 |
| 2.5 | 7.854 | 5.625 | 10.16 |
| 3 | 9.425 | 6.750 | 8.47 |
| 4 | 12.566 | 9.000 | 6.35 |
| 5 | 15.708 | 11.250 | 5.08 |
| 6 | 18.850 | 13.500 | 4.23 |
| 8 | 25.133 | 18.000 | 3.17 |
| 10 | 31.416 | 22.500 | 2.54 |
| 12 | 37.699 | 27.000 | 2.12 |
Series II, used when series I does not fit: 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5, 5.5, 7, 9, 11, 14, 18, 22, 28, 36, 45.
The module describes tooth size. It is the circular pitch divided by π, which is also the pitch diameter divided by the number of teeth. A module 2 gear has teeth twice as large as a module 1 gear.
Two gears mesh correctly only if they share the same module and the same pressure angle, so the module is the first thing to establish for a replacement gear. Imperial gears use diametral pitch instead: DP = 25.4 / m, so module 2.54 equals DP 10.
The pitch diameter is the circle on which two gears roll on each other without slipping; the centre distance comes from it. The tip (outside) diameter is what you measure with calipers across the tooth tips, and it is the diameter the blank is turned to. The root diameter is the bottom of the tooth spaces.
For the standard profile the addendum equals the module and the dedendum is 1.25 × module, hence da = d + 2m and df = d − 2.5m.
The gear ratio is the number of teeth on the driven gear divided by the number of teeth on the pinion: i = z2 / z1. The higher the ratio, the lower the output speed and the higher the torque, in the same proportion, minus mesh losses.
In a multi-stage gearbox the stage ratios multiply. Two stages of 4 give an overall ratio of 16.
The pressure angle is the inclination of the tooth flank at the pitch circle. Today's standard (ISO 53) is 20°. Older machines still have 14.5° gears, and some heavily loaded drives use 25°. Gears with different pressure angles do not mesh, even with the same module.
For helical gears the module is given in the plane normal to the tooth (the normal module, m), and the diameters use the transverse module mt = m / cos β. That is why a helical gear and a spur gear with the same module and tooth count have different diameters.
Helical teeth engage gradually, so the gear runs quieter and carries more load, but it creates an axial force the bearings have to take. The overlap ratio (εβ) grows with face width and helix angle.
On a pinion with too few teeth, the hob also cuts into the tooth root and weakens the tooth. For spur gears with a 20° pressure angle the theoretical limit is 17 teeth; in practice 14 works well. Below that, the gear is made with positive profile shift and the diameters and centre distance are recalculated.
The span over k teeth (W) is measured with a disc micrometer across the number of teeth the calculator gives. It checks tooth thickness independently of the tip diameter, which is why it is the measurement used on a gear hobbing machine between cuts.
The contact ratio (εα) shows how many tooth pairs are in contact on average during meshing. Below 1, the mesh is interrupted and the gears knock. In practice a contact ratio of at least 1.2 is the target; more teeth and helical teeth increase it.
For a spur gear with no profile shift, the module is the tip diameter divided by the number of teeth plus 2: m = da / (z + 2). Example: an 18-tooth gear with a 40 mm tip diameter has a module of 40 / 20 = 2. The calculator on this page does this and suggests the nearest standard module.
Divide the number of teeth on the driven gear by the number of teeth on the pinion: i = z2 / z1. An 18-tooth pinion driving a 45-tooth gear gives a ratio of 2.5, so the speed drops 2.5 times and the torque rises in almost the same proportion.
The centre distance is the average of the two pitch diameters: a = (d1 + d2) / 2 = m · (z1 + z2) / 2. For module 2 with 18 and 45 teeth it is 63 mm. For helical gears use the transverse module, m / cos β.
The module is 25.4 divided by the diametral pitch: m = 25.4 / DP. DP 10 equals module 2.54 and DP 8 equals module 3.175. Imperial and metric gears only mesh if the values match.
No. Two gears mesh correctly only with the same module and the same pressure angle. Small differences, such as module 2.5 against DP 10 (2.54), cause fast wear and noise.
Yes. Enter the helix angle β and the calculator switches to the transverse module for the diameters, computes the overlap ratio from the face width, the virtual number of teeth and the helix lead.
They are nominal values for the ISO 53 basic rack and gears with no profile shift. For manufacturing, add the tolerances of the accuracy grade and the backlash; profile-shifted gears are recalculated with the shift coefficient.
The calculation uses the ISO 53 basic rack (addendum equal to the module, bottom clearance 0.25 × module) and gears with no profile shift. Profile-shifted gears have different diameters and centre distance.
Diameters and span measurement are nominal. Backlash and allowable deviations depend on the accuracy grade of the gear pair and are subtracted from the span measurement.
The tip diameter of a worn or profile-shifted gear differs from the nominal one. The result points to the nearest standard module; confirm it with a span measurement or a module gauge.
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