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MachineCalcs

Gear Tooth Ratio Calculator

Gear tooth ratio for a single or two-stage (compound) gear train — each stage is driven ÷ driving teeth, and the stages multiply for the overall ratio. The ratio is unit-independent; output speed is shown in RPM.

Inputs

rpm

Results

Total ratio(i) ✓
3

3.00:1 reduction (output slower)

Stage 1 × stage 2 (n:1).

Stage 1 ratio(i₁)
3

z₂ ÷ z₁.

Stage 2 ratio(i₂)
1

z₄ ÷ z₃ (1 if single stage).

Output speed(n₂)
500 rpm

  • Each stage ratio is driven teeth ÷ driving teeth; the stages multiply for the total compound ratio.
  • Single stage: leave the stage 2 teeth at 0.
  • An idler gear between two gears changes the direction of rotation only — it does not change the ratio.
  • Ratios are unit-independent and assume external spur meshes (each stage reverses rotation); efficiency losses are ignored.

How it works

The gear tooth ratio of one mesh is the driven tooth count divided by the driving tooth count: i = z₂ / z₁ A ratio above 1:1 is a reduction — the output turns slower than the input. For a gear train of several meshes in series, the overall (compound) ratio is the product of the stage ratios: i = (z₂/z₁) · (z₄/z₃). The output speed is the input speed divided by the total ratio, n₂ = n₁ / i.

An idler gear placed between two gears only reverses the direction of rotation — it cancels out of the ratio and does not change it. Each external spur mesh reverses rotation, so the output of a two-stage train turns the same way as the input.

Worked example

A 12-tooth gear driving a 36-tooth gear is 36 ÷ 12 = 3, a 3:1 reduction; at 1,500 RPM in, the output turns 500 RPM. Add a second stage of a 15-tooth gear driving a 45-tooth gear (another 3:1) and the train becomes 3 × 3 = 9, a 9:1 overall ratio, dropping 1,500 RPM to ≈ 167 RPM. Those are the numbers the calculator returns for these inputs.

Frequently asked questions

How do you calculate gear tooth ratio?
Divide the driven (output) tooth count by the driving (input) tooth count for each mesh: ratio = z₂ / z₁. For 12 driving teeth and 36 driven teeth that is 36 ÷ 12 = 3, a 3:1 reduction.
What is a gear train and how do I find the compound ratio?
A gear train is two or more meshes in series. Find each stage ratio (driven ÷ driving), then multiply the stages for the overall (compound) ratio: i = i₁ · i₂. A 3:1 stage followed by another 3:1 stage gives 9:1 overall.
Does an idler gear change the gear ratio?
No. An idler gear sits between the driving and driven gears and only reverses the direction of rotation — it cancels out of the ratio, so the overall ratio depends only on the first driving and last driven gears.
What is the difference between a reduction and an overdrive ratio?
A reduction (ratio > 1, e.g. 3:1) turns the output slower than the input but with proportionally more torque. An overdrive (ratio < 1) turns the output faster with less torque. The total ratio sets the output speed: n_out = n_in ÷ total ratio.
How is this different from a gear ratio calculator?
This solves the pure tooth-count ratio of a single or compound gear train and the output speed. If you need the resulting torque and vehicle (road) speed from tire size, use the gear ratio calculator instead.
Does the gear tooth ratio depend on metric or imperial units?
No — the tooth ratio is just a count divided by a count, so it is unit-independent and the same in any system. Only the speed result carries a unit (RPM).

Method & assumptions

  • External spur meshes — each stage reverses the direction of rotation.
  • Idler gears are omitted from the ratio; they affect direction only.
  • The tooth ratio is exact and unit-independent; this ignores efficiency (mesh) losses.

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