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GearLab gears & robot gearboxes
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2. Module, teeth and the involute

Two numbers describe almost everything about a gear: the module m, which is the size of a tooth, and the number of teeth z, which is the size of the wheel. Their product is the reference diameter.

d = m · z     p = π · m     a = m (z1 + z2)2KHK

Everything else on a standard tooth follows from the module:

ha = m  ·  hf = 1.25 m  ·  da = d + 2m  ·  df = d − 2.5m  ·  db = d · cos αKHK

Two gears only mesh if they share a module and a pressure angle. That is the whole reason modules are standardised — a cutter is ground for one module, and it cuts every tooth count.

base circle d_b Unwind a taut string from the base circle. The path of its end is the involute. That single property is why the ratio stays constant even if the centre distance is off.
The involute is the only tooth shape whose contact point runs along a straight line — the line of action. Move the wheels apart and the pressure angle changes, but the speed ratio does not.

Why the involute, and not some other curve

The involute has a property no other practical curve has: as the wheels turn, the contact point travels along a straight line, inclined at the pressure angle α. Two consequences follow, and both are why every power gear on earth is an involute:

  • The speed ratio is exactly z₂/z₁ at every instant, not on average. A non-involute tooth would make the output speed ripple within each tooth pitch.
  • Move the wheels apart and it still works. The pressure angle opens up and the contact ratio drops, but the ratio itself does not change. This is what makes the centre distance a tolerance rather than a fit.

The mathematics of the curve is one function, which you will meet again in the next chapter:

inv α = tan α − α     inv 20° = 0.014904
Pressure angle. 20° is the modern default. A larger angle (25°) gives a thicker, stronger root and tolerates fewer teeth, at the cost of a lower contact ratio and higher bearing loads. 14.5° was the old standard and survives mostly in legacy repairs.
Try it

A 2-module pair, 20 teeth driving 40. Reference diameters 40 and 80 mm, centre distance 60 mm, ratio 2:1.

Formulas in this chapter

d — reference diameter
d = m · z [mm] KHK Gears
d_a, d_f, d_b — tip, root and base diameter
d_a = d + 2m, d_f = d − 2.5m, d_b = d cos α [mm] KHK Gears
a — reference centre distance
a = m (z₁ + z₂) / 2 [mm] KHK Gears
inv α — involute function
inv α = tan α − α; inv 20° = 0.014904 [rad] involute geometry
p — circular pitch
p = π m [mm] definition