Gear Ratio Calculator
Work out gear ratio from tooth counts, with output speed, torque multiplication, and multi-stage or compound gear trains.
How to use this calculator
- 1Enter the tooth counts for the driving (input) and driven (output) gears.
- 2Add a second stage for a compound train — the ratios multiply.
- 3Enter input speed and torque to see what comes out the other end.
How the calculation works
ratio = driven teeth ÷ driving teeth output rpm = input rpm ÷ ratio output torque = input torque × ratio compound: ratio = ratio₁ × ratio₂- driving gear
- The input gear, turned by the motor, engine or pedals
- driven gear
- The output gear, which turns whatever the machine drives
- ratio > 1
- Reduction — slower output, more torque
- ratio < 1
- Overdrive — faster output, less torque
Speed and torque trade off exactly. A 3:1 reduction gives a third of the speed and three times the torque, because gears redistribute power rather than creating it — power in equals power out, minus friction.
Compound gear trains multiply. Two 6:1 stages give 36:1, not 12:1, which is how a gearbox achieves a large reduction without a gear the size of a wheel.
Tooth counts sharing no common factor mean every tooth eventually meets every other tooth, spreading wear evenly. Designers add a "hunting tooth" for exactly this reason — 12 and 36 share a factor of 12 and wear unevenly, while 13 and 36 do not.
Idler gears between the driving and driven gear do not change the ratio at all. They only reverse the direction of rotation and bridge distance.
Worked example
A 3:1 reduction from a 1800 rpm motor
- 1.Ratio: 36 driven ÷ 12 driving = 3:1.
- 2.Output speed: 1800 ÷ 3 = 600 rpm.
- 3.Output torque: 10 × 3 = 30 N·m.
- 4.Power is unchanged: speed fell threefold and torque rose threefold.
- 5.12 and 36 share a common factor of 12, so the same teeth mesh repeatedly — a 13-tooth driver would spread wear far better.
Result: 3:1 — 600 rpm, 30 N·m
Two stages multiply
- 1.Stage 1: 36 ÷ 12 = 3:1, taking 1800 rpm down to 600.
- 2.Stage 2: 42 ÷ 14 = 3:1, taking 600 rpm down to 200.
- 3.Overall: 3 × 3 = 9:1, not 3 + 3 = 6:1.
- 4.Output torque: 10 × 9 = 90 N·m.
- 5.A single-stage 9:1 would need a 108-tooth gear against the 12-tooth driver — far larger than two small stages.
Result: 9:1 overall — 200 rpm, 90 N·m
Gears trade speed for torque and nothing else
A gear ratio does not create power. Power is torque multiplied by rotational speed, and a gear train keeps that product roughly constant — so anything gained in torque is lost in speed, exactly in proportion. A 10:1 reduction gives ten times the torque at a tenth of the speed.
This is why a bicycle in a low gear climbs a hill easily but slowly, and why an electric drill has a gearbox: the motor produces modest torque at high speed, and the drill needs the reverse. The gearbox converts one into the other.
The only thing lost in the exchange is efficiency. A well-made spur gear mesh runs at 97–99%, so a two-stage box still delivers around 95% of the power in. Worm drives are far worse — often under 50% — which is the price paid for their large ratio in a single compact stage and their resistance to being back-driven.
The hunting tooth
If the driving and driven tooth counts share a common factor, the same pairs of teeth meet over and over. A 12-tooth gear driving a 36-tooth gear means each driver tooth only ever contacts three of the driven teeth, forever.
That concentrates wear. Any imperfection on one tooth — a machining mark, a hard spot, a slight profile error — grinds against the same three partners for the life of the gearset, and the damage compounds.
Making the counts coprime fixes it. Change the driver to 13 teeth and every tooth eventually meets every one of the 36, spreading wear and any defect across the whole gear. This deliberately added tooth is called a hunting tooth, and it is why production gearboxes so often use odd counts like 13, 17 or 19 where a round number would have been easier to design with.
What this assumes, and where it stops
Assumptions
- Ideal gears with no friction, backlash or losses.
- Ratio is defined as driven teeth divided by driving teeth, so values above 1 are reductions.
- Compound stages are fixed-axis, so their ratios multiply.
Limitations
- Ignores efficiency. Real spur meshes lose 1–3% per stage and worm drives can lose more than half, so actual output torque is lower.
- Fixed-axis trains only. Planetary and epicyclic gearsets have a different relationship depending on which element is held.
- Does not check whether the gears physically mesh — that requires matching module or diametral pitch, which is not modelled.
- Says nothing about tooth strength, bearing loads or lubrication, all of which govern whether a ratio is practical.
Common questions
How do you calculate gear ratio?
Divide the driven gear's tooth count by the driving gear's. A 36-tooth gear driven by a 12-tooth gear gives 36 ÷ 12 = 3:1. Values above 1 are reductions, giving less speed and more torque; values below 1 are overdrives, giving the reverse.
Do gear ratios add or multiply in a gearbox?
They multiply. Two 3:1 stages give 9:1 overall, not 6:1. This is exactly why multi-stage gearboxes exist — achieving 9:1 in one stage would need a driven gear nine times the diameter of the driver, while two small stages fit in a fraction of the space.
Does an idler gear change the ratio?
No. A gear placed between the driving and driven gears reverses the direction of rotation and bridges the distance between shafts, but the overall ratio still depends only on the first and last gears. Its own tooth count cancels out entirely.
Why do gearboxes use odd tooth counts?
To make the counts share no common factor, so every tooth eventually meets every tooth on the mating gear. This spreads wear evenly. If 12 and 36 mesh, each driver tooth only ever touches three of the driven teeth, concentrating any defect on the same partners for the life of the gearset.
Sources
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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