About Gear Ratio Calculator — RPM, Torque & Compound Trains
The gear ratio calculator turns tooth counts into speed and torque. Enter the driver (input) and driven (output) tooth counts of each meshing pair — one pair for a simple train, up to six for a compound train — and the tool computes each stage ratio (driven ÷ driver), multiplies them into the overall ratio, and converts your input speed and torque to the output shaft: RPM out = RPM in ÷ ratio, torque out = torque in × ratio × efficiency.
A ratio above 1 is a reduction — the output turns slower but harder, which is why a 3:1 gearbox turns 10 N·m into nearly 30. A ratio below 1 is an overdrive that trades torque for speed. Compound trains exist because big single-stage ratios need impractically large gears: two modest 3:1 stages give 9:1 in a fraction of the space one 9:1 pair would need.
How It Works
- Enter the tooth counts of each gear pair in power-flow order: the driver is the gear receiving power, the driven is the gear it meshes with. Tooth counts are whole numbers — the calculator rejects fractional teeth because they cannot exist on a physical gear.
- Add stages for a compound train. In a compound train each intermediate shaft carries two gears (the driven gear of one stage and the driver of the next), and the overall ratio is the product of the stage ratios: (60/20) × (45/15) = 3 × 3 = 9:1.
- Enter the input speed and torque, and an efficiency if you want realistic output torque — a well-lubricated spur or helical stage transmits roughly 97–99% of its power, so a two-stage box is commonly modeled near 0.95–0.97 overall. Leave efficiency at 1.00 for the ideal kinematic answer.
- Read the results: overall ratio, output RPM (input ÷ ratio — efficiency does not change speed, only torque), output torque (input × ratio × η), and the torque multiplication factor ratio × η.
Worked Example
A 1800 RPM motor drives a two-stage reduction: a 20-tooth pinion on the motor meshing with a 60-tooth gear (3:1), then a 15-tooth pinion on the intermediate shaft meshing with a 45-tooth gear (another 3:1). The overall ratio is 3 × 3 = 9:1, so the output shaft turns 1800 ÷ 9 = 200 RPM. With 10 N·m at the motor and a 95% train efficiency, the output torque is 10 × 9 × 0.95 = 85.5 N·m — the speed dropped 9-fold and the torque rose almost 9-fold, with the missing 5% lost to friction. The chart below shows the same 1800 RPM input through common single-stage ratios at ideal efficiency.
Gear ratio RPM chart: 1800 RPM input through common ratios
Output speed and torque multiplication for a 1800 RPM input shaft — the 4-pole induction-motor speed — through common reductions at ideal efficiency (η = 1.00; enter your own efficiency in the tool to include losses). The 9:1 row is a two-stage compound train; the rest are single pairs.
| Ratio (teeth) | Output speed (RPM) | Torque multiplier |
|---|---|---|
| 2:1 (20T → 40T) | 900 | 2.00× |
| 3:1 (20T → 60T) | 600 | 3.00× |
| 4:1 (20T → 80T) | 450 | 4.00× |
| 5:1 (20T → 100T) | 360 | 5.00× |
| 9:1 compound (20:60 × 15:45) | 200 | 9.00× |
Formulas
- Stage ratio
e = N_driven / N_driver- Compound train ratio
e_total = e₁ × e₂ × … × eₙ- Output speed
RPM_out = RPM_in / e_total- Output torque
T_out = T_in × e_total × η
Standards & References
- Gear-train kinematics per standard machine-design references (e.g. Shigley's Mechanical Engineering Design, gear-train chapters): train value as the product of tooth-count ratios, speed inversely and torque directly proportional to the ratio
- Torque relation from power conservation: P_out = P_in × η and T = P/ω, giving T_out = T_in × e × η
- Typical per-stage efficiency for enclosed spur/helical gearing ~97–99%; worm and high-ratio single stages run far lower — use the manufacturer's figure when you have one
Frequently Asked Questions
How do I calculate a gear ratio from tooth counts?
Divide the driven gear's tooth count by the driver's: a 20-tooth pinion driving a 60-tooth gear is 60 ÷ 20 = 3:1. The output shaft turns 3× slower and, ignoring losses, carries 3× the torque. Tooth counts work because meshing gears must pass teeth at the same rate — the ratio is exact, unlike measuring diameters.
How does a compound gear train multiply ratios?
In a compound train each intermediate shaft carries two gears: the driven gear of one stage and the driver of the next. Because both turn at the same shaft speed, the stage ratios multiply — (60/20) × (45/15) = 9:1 in the worked example. That is how gearboxes reach large reductions with small gears: a single 9:1 pair would need a 180-tooth gear against a 20-tooth pinion.
Does a gear reduction really increase torque?
Yes — torque scales with the ratio, minus friction. Power is torque × angular speed, and power is (nearly) conserved through the train, so when a 3:1 reduction cuts the speed to a third, the torque rises to 3 × η times the input. The calculator's torque multiplier is exactly ratio × efficiency: 2.85× for 3:1 at 95%.
What efficiency should I use?
Enclosed, lubricated spur and helical stages typically transmit 97–99% of their power each, so a two-stage box lands near 0.95–0.97 overall. Use 1.00 when you only need the kinematics (speeds always follow the tooth counts exactly), and the manufacturer's figure for anything with worm gears — a worm stage can drop below 70% and the ratio × η shortcut still applies.
What is the difference between a reduction and an overdrive?
A reduction has ratio > 1 (small gear drives big gear): slower and stronger output — the normal case for motors, which make power at high RPM. An overdrive has ratio < 1 (big drives small): faster but weaker output, used where the load needs speed, like a bicycle's big chainring driving a small cog. The calculator accepts both; enter the driver and driven teeth as physically arranged.
Why does the calculator reject fractional tooth counts?
A physical gear has a whole number of teeth, and the ratio between meshing gears is exactly the integer tooth-count ratio — that exactness is the reason gears (not belts) drive camshafts and clocks. If you are back-calculating from a measured ratio, find the nearest integer pair: 2.95:1 is likely a 59/20 pair, not a rounding error.