Spur gear calculator: pitch, tip and root diameter from module and teeth
This spur gear calculator gives the dimensions of standard spur gears (no profile shift) from the module m and the number of teeth z, with the formulas of the KHK gear technical reference: reference diameter d = z × m, tip diameter d + 2m, root diameter d − 2.5m. KHK’s own example – module 3, 12 and 24 teeth – gives a center distance of 54 mm.
Example: Calculate: gear dimensions from module and teeth · Module m 3 mm · Teeth pinion z1 12 · Teeth gear z2 24 · Pressure angle α 20 ° → Result: Center distance 54.000 mm – ratio i = 2. Source: KHK Kohara Gear – Gear Technical Reference, 4.1 Spur Gears, Table 4.1 “Calculations for Standard Spur Gears” and Table 4.2 “Calculations for Number of Teeth” (read 09.10.2026). Updated: .
How it is calculated
The module is the size of the tooth. KHK defines the circular pitch as the pitch circle circumference divided by the number of teeth, p = π × d ÷ z. Dividing out π gives the module m = d ÷ z – a clean number such as 1, 2 or 3 mm. Two gears only mesh when their module (and pressure angle) are the same.
Formulas for standard spur gears
- Reference diameter d = z × m
- Tip (outside) diameter da = d + 2m, root diameter df = d − 2.5m
- Addendum ha = 1.00m, tooth depth h = 2.25m
- Base diameter db = d × cos α
- Center distance a = (z1 + z2) × m ÷ 2
Examples
| Module | Teeth | d | da | df |
|---|---|---|---|---|
| 1 | 20 | 20 mm | 22 mm | 17.5 mm |
| 2 | 30 | 60 mm | 64 mm | 55 mm |
| 2.5 | 40 | 100 mm | 105 mm | 93.75 mm |
| 3 | 12 | 36 mm | 42 mm | 28.5 mm |
| 3 | 24 | 72 mm | 78 mm | 64.5 mm |
In KHK’s example (module 3, α = 20°, 12 and 24 teeth) the base diameters are 33.829 and 67.658 mm, the tooth depth 6.75 mm and the center distance 54 mm. The circular pitch at module 3 is 9.425 mm.
Teeth from center distance and ratio
If the center distance a and the speed ratio i are fixed, the sum of teeth is 2a ÷ m. KHK: module 3, a = 54 mm and i = 1.25 give 36 teeth, split into 16 and 20. If the result is not a whole number – module 2, a = 50 mm, i = 3 gives 12.5 and 37.5 – the gears need profile shift or a helix. For speeds and ratios of whole gear trains use the gear ratio calculator; circle values are in the circle calculator.
Frequently asked questions
How do you calculate the pitch diameter of a gear?
Number of teeth times module: 24 teeth at module 3 give d = 72 mm.
How do I find the outside diameter?
Tip diameter da = d + 2m. A 20-tooth gear with module 1 measures 22 mm across the tips.
What is the root diameter?
df = d − 2.5m. For 30 teeth at module 2: 60 − 5 = 55 mm.
How do you calculate the module from a gear?
Measure the tip diameter and count the teeth: m = da ÷ (z + 2). 64 mm and 30 teeth give module 2.
How is the center distance calculated?
a = (z1 + z2) × m ÷ 2. Module 3 with 12 and 24 teeth: 36 × 3 ÷ 2 = 54 mm.
What is the circular pitch?
The tooth spacing on the pitch circle, p = π × m. At module 3 it is 9.425 mm.
Sources and legal basis
- KHK Kohara Gear – Gear Technical Reference, 4.1 Spur Gears, Table 4.1 “Calculations for Standard Spur Gears” and Table 4.2 “Calculations for Number of Teeth” (read 09.10.2026)
- KHK Kohara Gear – Gear Nomenclature “Circular pitch”: p = πd ÷ z, module m = d ÷ z (read 09.10.2026)
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