The module: Pitch in a Single Number
The module m is the key parameter of any gear. It describes the ratio of the pitch diameter d to the number of teeth z and is measured in millimeters:
m = d / z
m = module (mm) · d = pitch diameter (mm) · z = number of teeth
Directly linked to the module is the pitch p—the distance between two consecutive tooth flanks on the reference circle:
p = π · m
Two gears mesh only if they have the same pitch—that is, the same module.
Reference circle, top circle, and bottom circle
All diameters of a standard tooth profile (without profile shift, 20° pressure angle) can be derived from the module and the number of teeth:
- Reference circle d = m · z – the reference circle on which the pitch p lies
- Pitch circle dₐ = d + 2 · m = m · (z + 2) – outer diameter; this is where the tooth tips are located
- Root circle d_f = d − 2.5 · m = m · (z − 2.5) – base of the tooth root
The reference circle is a fixed, purely geometric property of a single wheel—it does not change when the wheel is paired with another wheel. This distinguishes it from the operating pitch circle, which is created only through the interaction of two wheels (more on this in the next unit).
Center distance of a gear pair
For two meshing gears with the same module m and the same number of teeth z₁ and z₂, the center distance is given by the sum of half the pitch diameters:
a = m · (z₁ + z₂) / 2
a = center distance (mm) · z₁, z₂ = number of teeth on the two gears
DIN 780: Why Standard Modules Are Important
DIN 780 specifies preferred module values to ensure that gears from different manufacturers remain interchangeable and to simplify inventory management:
- Series 1 (preferred): 1 · 1.25 · 1.5 · 2 · 2.5 · 3 · 4 · 5 · 6 · 8 · 10 · 12 · 16 · 20 mm
- Series 2 (alternative): 1.125 · 1.375 · 1.75 · 2.25 · 2.75 · 3.5 · 4.5 · 5.5 · 7 · 9 · 11 · 14 · 18 mm – only if Series 1 does not fit
Mnemonic
Worked example
Given
A gear has a module of m = 4 mm and a number of teeth of z = 18.
Calculation
- Reference circle: d = m · z = 4 · 18 = 72 mm
- Pitch circle: dₐ = d + 2 · m = 72 + 8 = 80 mm
- Circle of diameter: d_f = d − 2.5 · m = 72 − 10 = 62 mm
- Pitch: p = π · m = π · 4 ≈ 12.57 mm
If this gear (z₁ = 18) meshes with a second gear of the same module and z₂ = 30, the center distance is a = 4 · (18 + 30) / 2 = 96 mm.
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