Gear Ratio Calculator
Enter driver and driven tooth counts for each stage to get the overall gear ratio, output rpm, output torque after mesh losses and the direction of rotation.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Gear Ratio Calculator?
A gear ratio tells you how many turns the input makes for one turn of the output. A 12-tooth pinion driving a 36-tooth gear is a 3 : 1 reduction: the output turns a third as fast and, ignoring friction, with three times the torque. Put two such stages on intermediate shafts and the ratios multiply, so 12 : 36 followed by 15 : 45 gives 9 : 1. This calculator handles that bookkeeping for up to eight stages.
Each stage is one mesh between a driver and a driven gear. You can add idlers between them, which change the direction of rotation without changing the ratio, and mark a stage as an internal ring gear, which turns the same way as its pinion. Enter the input speed and torque and the result shows output rpm, output torque with a per-mesh efficiency applied, the power lost as heat and which way the output shaft turns.
A stage-by-stage table lists the speed and torque on every intermediate shaft, which is what you need to size the shafts, keys and bearings in the middle of a train rather than just the ends.
- Up to eight stages with idlers and internal ring gears
- Overall ratio shown as reduction (n : 1) or overdrive (1 : n)
- Output speed, torque with compounding mesh losses, power lost and direction of rotation
- Stage-by-stage speed and torque for every intermediate shaft
- Torque in N·m, lb·ft, lb·in or kgf·m; copy and share link
How to use the Gear Ratio Calculator
- For stage 1, enter the teeth on the driving gear and on the gear it drives. Add idlers if there are gears between them on separate shafts.
- If the driven gear is an internal ring gear, choose Internal (ring) gear for that stage.
- Click Add stage for each further mesh in a compound train. The driven gear of one stage and the driver of the next share a shaft.
- Enter the input speed, the input torque with its unit, and the efficiency per mesh (98% is a reasonable default for spur gears).
- Read the overall ratio, output speed, output torque and direction, then check the stage table for intermediate shafts.
Worked examples
Two-stage compound reducer (page default)
Input
12 → 36, then 15 → 45; 1,750 rpm, 20 N·m in, 98% per mesh
Output
9 : 1; 194.4 rpm; 172.9 N·m; same direction as input
Lossless output would be 180 N·m. Two meshes at 98% leave 96.04%, so 7.1 N·m is lost to friction. Two external meshes reverse twice, so the output turns the same way as the motor.
Adding an idler
Input
20 → 60 with 1 idler; 1,500 rpm, 10 N·m, 98%
Output
3 : 1; 500 rpm; 28.81 N·m; same direction
Without the idler the output would turn the opposite way at the same 500 rpm. The idler costs a second mesh of efficiency, which is the price of reversing direction.
Pinion driving a ring gear
Input
18 → 72 internal; 1,200 rpm, 10 N·m, 98%
Output
4 : 1; 300 rpm; 39.2 N·m; same direction
An internal mesh gives a reduction in a compact space without reversing rotation.
Background and insight
For two meshing gears the pitch-line speeds are equal, so N1 x n1 = N2 x n2, where N is the number of teeth and n the speed. The speed ratio of one mesh is therefore driven teeth divided by driver teeth. In a compound train each intermediate shaft carries the driven gear of one mesh and the driver of the next, so the overall ratio is the product of the stage ratios. An idler gear appears once as a driven gear and once as a driver, so its tooth count cancels.
Torque scales the other way. Power is torque times speed, and a gear mesh only loses a small fraction of it to friction, so output torque is input torque times the ratio times the efficiency. Well-made spur and helical gears run at about 98 to 99% per mesh; worm gears are far lower and are not modelled here. Losses compound: four meshes at 98% leave 92% of the power.
Direction follows a simple rule. Every external mesh reverses rotation, so an even number of external meshes returns the original direction. A pinion inside an internal ring gear turns the same way as the ring, so that mesh does not reverse. Planetary gearsets combine these and need their own formula (the Willis equation), which this calculator does not attempt.
Common use cases
Designing a reducer
Split a required ratio into stages, see the speed and torque on every shaft and check the total mesh loss.
Robotics and hobby builds
Work out output speed and stall torque for a small DC motor driving through a printed or off-the-shelf gear train.
Clocks, models and mechanisms
Check that a chain of gears gives an exact ratio and the direction you need before cutting any metal.
Machine retrofits
Change one gear in an existing train and see how the output speed and torque change.
Tips and limitations
- Avoid a pinion below about 17 teeth at a 20 degree pressure angle unless the gears are profile-shifted; fewer teeth risk undercut.
- Choose tooth counts that share no common factor (a hunting tooth ratio) so the same teeth do not meet every revolution and wear evenly.
- Keep single-stage ratios under about 6 : 1 for spur gears. Larger reductions are usually cheaper and smaller as two stages.
- Efficiency here is per mesh at full load. Light loads, splash lubrication and seals can make real efficiency lower.
- Torque on the output shaft rises with each stage. The last stage needs the widest face and the strongest teeth.
Frequently asked questions
How do you calculate the gear ratio of two gears?
Divide the teeth on the driven gear by the teeth on the driving gear. A 15-tooth driver and 45-tooth driven gear give 45 / 15 = 3, a 3 : 1 reduction: the output turns at one third of the input speed.
How do you calculate a compound gear ratio?
Multiply the ratios of each mesh. For 12 : 36 followed by 15 : 45 the overall ratio is 3 x 3 = 9 : 1. Gears sharing a shaft turn at the same speed, which is why their ratios multiply.
Does an idler gear change the gear ratio?
No. Its teeth cancel out, because it is a driven gear for one mesh and a driver for the next. It does reverse the direction of rotation and adds one mesh of friction loss.
How does gear ratio affect torque?
Output torque equals input torque times the ratio, less friction. A 9 : 1 reduction with 20 N·m in gives 180 N·m out if lossless, or about 173 N·m with two meshes at 98% efficiency.
Which way does the output gear turn?
Each external mesh reverses rotation. One mesh: opposite. Two meshes or one idler: same direction. A pinion driving an internal ring gear does not reverse.
What efficiency should I use for spur gears?
About 98 to 99% per mesh for well-lubricated, accurately cut spur or helical gears at rated load. Use lower values for plastic gears, light loads or poor lubrication. Worm drives can be 50 to 90% and need a separate calculation.
Is a gear ratio of 4:1 the same as 0.25?
They describe the same pair from different ends. 4 : 1 is the speed ratio as a reduction (input over output). Expressed as output over input it is 0.25. This calculator always reports input speed divided by output speed.
Is there an in-depth guide to the Gear Ratio Calculator?
Yes. “Gear Train Ratios Explained: Simple, Compound, Idlers and Losses” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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