The gear ratio calculator above handles a train rather than a single pair, because almost every real gearbox has more than one stage and the stages multiply. Enter tooth counts or pitch diameters for up to three meshes, add an input speed and torque, and it returns the overall ratio together with what comes out the far end. It also applies a per-stage efficiency, so the torque figure reflects the losses that a purely ideal calculation quietly ignores.
Arb Digital builds free calculators that show the trade rather than just the number. A gear ratio is not a free multiplication: torque up means speed down by the same factor, and the product of the two — power — can only ever decrease. This page also states the boundary against the site's converters, which rescale a torque figure between units rather than deriving a new one from a gear train.
What This Gear Ratio Calculator Does
It computes the overall ratio of a gear train as the product of the individual stage ratios, where each stage ratio is driven teeth divided by driver teeth. A ratio above 1 is a reduction: the output turns slower than the input and with more torque. A ratio below 1 is an overdrive, turning faster with less torque. The hero shows the ratio in the conventional colon form.
The grid gives output speed in revolutions per minute, output torque in newton metres, the same torque in pound-feet for imperial workflows, and the direction the output turns relative to the input. That last item is genuinely useful and often forgotten: each external mesh reverses direction, so an odd number of stages turns the output backwards while an even number restores the original sense.
Efficiency is applied per stage and compounds. At 97 per cent each, two stages deliver 94.1 per cent and three deliver 91.3 per cent. That figure multiplies the torque output, not the speed, because speed is fixed by geometry while losses show up as heat taken out of the torque.
How to Use It
- Choose tooth counts or diameters. Either produces the same ratio for meshing gears, since a common tooth pitch means diameter is proportional to tooth count.
- Fill in as many stages as you have. Leave the unused stages at zero and they are skipped. A single pair uses stage one only.
- Enter the input speed and torque. These are the values at the driving shaft — motor speed and motor torque, or engine speed and engine torque.
- Set a realistic efficiency. Use 100 per cent for a textbook answer, or a figure from the gearbox manufacturer for a practical one. Worm drives are far less efficient than spur gears.
- Check the direction indicator. If your machine needs a particular rotation sense, an odd number of external meshes will surprise you.
The Formula: How Gear Ratio Is Calculated
For one mesh, the gear ratio is the number of teeth on the driven gear divided by the number on the driver. Output speed is input speed divided by that ratio, and output torque is input torque multiplied by it. The Duke MEMS gear train reference states the same three relations directly: the ratio is output teeth over input teeth, the following gear's speed is found by dividing by the gear ratio, and the output torque is found by multiplying the input torque by it.
For a train, ratios multiply: Rtotal = R1 × R2 × R3. Work the defaults. Stage one is 12 teeth driving 36, a ratio of 3. Stage two is 15 driving 45, also 3. Overall the ratio is 9:1. An input of 1,800 rpm becomes 1,800 ÷ 9 = 200 rpm at the output.
Torque goes the other way. An input torque of 10 N·m becomes 10 × 9 = 90 N·m in the ideal case. Applying 97 per cent efficiency to each of the two stages gives 90 × 0.97 × 0.97 = 84.68 N·m, which is 62.46 lb·ft. The newton metre used here is the coherent SI unit for torque, built from the newton and the metre as set out in the SI Brochure published by the BIPM.
Gears Trade Speed for Torque and Never Create Power
Mechanical power is torque multiplied by angular speed. If a gear train multiplies torque by nine and divides speed by nine, the product is unchanged — power in equals power out, in the ideal case. This is not a coincidence or an approximation; it is conservation of energy, and no arrangement of gears can get around it.
Real trains do slightly worse. Friction at the tooth flanks, churning of the lubricant and bearing drag all remove energy, which appears as heat. The output power is therefore input power times the efficiency, and since speed is set rigidly by the tooth counts, the entire loss shows up as reduced torque. That is why the efficiency field in this tool multiplies the torque and leaves the speed alone.
The practical implication is that a gearbox is a matching device, not an amplifier. An electric motor produces its best efficiency over a narrow speed band; the load may need a completely different speed. The gearbox reconciles the two without changing how much power is available, and choosing the ratio is really choosing where on the motor's curve you want to operate. The electrical power calculator gives the electrical input side of that picture.
Idler Gears, Direction and Compound Trains
An idler gear sits between a driver and a driven gear and meshes with both. It has no effect on the overall ratio whatsoever, because its tooth count appears once in a numerator and once in a denominator and cancels. What it does change is direction: each external mesh reverses rotation, so adding an idler flips the output sense while leaving the speed and torque untouched.
This tool treats each stage as one mesh and reports the resulting direction. If your train includes idlers, count them as stages with equal driver and driven values, which gives a ratio of 1 and correctly flips the direction indicator. Internal gears, where a pinion runs inside a ring gear, do not reverse direction, so a train using them needs the sense checked by hand.
A compound train is one where two gears share a shaft, so the second stage begins at the speed the first stage ended. That is exactly what the multi-stage fields here describe, and it is how large ratios are built without absurdly large gears. Achieving 9:1 in a single mesh would need a driven gear nine times the diameter of the driver; two 3:1 stages achieve the same thing in a fraction of the space.
Choosing a Ratio for a Real Machine
Ratio selection almost always starts from the load, not the motor. Work out the speed the output needs to turn and the torque it needs to deliver, then divide backwards to find what the input must supply. If the required input torque exceeds what your motor produces, the ratio is too low; if the required input speed exceeds what the motor can reach, the ratio is too high.
Two constraints then narrow the choice. Very high single-stage ratios are impractical because the driven gear grows physically large, which is why multi-stage trains exist. And very low tooth counts on the driver cause undercutting and weak teeth, so pinions below about 17 teeth need profile correction in standard involute systems. Between those limits, the ratio is a design choice rather than a calculation.
Where Gear Efficiency Actually Goes
The efficiency field is a single number standing in for several distinct loss mechanisms. Sliding friction between tooth flanks is the largest for most gear types, and it depends on how much the teeth slide relative to each other as they roll through the mesh. Involute spur and helical gears are designed to minimise that sliding, which is why they routinely reach 97 to 99 per cent per stage.
Worm drives are the opposite case. The worm slides along the wheel rather than rolling on it, so a worm stage may run at 50 to 90 per cent depending on lead angle and lubrication. That inefficiency is sometimes wanted rather than tolerated, because a sufficiently inefficient worm drive cannot be back-driven, which makes it self-locking under load. If you are modelling a worm stage here, set the efficiency accordingly rather than leaving it near 100.
The remaining losses are churning of lubricant, windage at high speed, and bearing and seal drag. These are largely speed-dependent rather than load-dependent, which means efficiency is not constant across a gearbox's operating range: a lightly loaded gearbox at high speed can be markedly less efficient in percentage terms than the same box working hard at moderate speed. Manufacturer efficiency figures usually assume rated load.
How This Differs From the Torque Converter
The boundary in one sentence: a converter rescales an existing torque between units, while this calculator derives a new torque from a gear train. The torque converter turns newton metres into pound-feet or kilogram-force metres; it knows nothing about tooth counts. This page produces an output torque that did not exist as an input, by applying the gear ratio to what you supplied.
The same distinction runs through the related tools. The power converter and watts to horsepower converter rescale power figures, and the speed converter handles linear speeds if you are working out a road speed from a wheel diameter. Calculators including the angular velocity calculator, the force calculator and the general ratio calculator derive quantities, and each covers a different piece of a drivetrain problem.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Inverting the ratio — driven teeth divided by driver teeth gives a reduction above 1. Getting it upside down turns a reduction into an overdrive.
- Counting an idler in the ratio — it cancels out completely and changes only the direction of rotation.
- Adding stage ratios instead of multiplying — two 3:1 stages give 9:1, not 6:1.
- Expecting a gearbox to create power — torque multiplication always costs exactly the same factor in speed, and efficiency losses take a little more.
- Using outside diameter instead of pitch diameter — the ratio depends on the pitch circles where the teeth actually mesh, not on the outer tips.
Related Free Tools From Arb Digital
Rescale results with the torque converter, the power converter or the watts to horsepower converter. For rotational motion use the angular velocity calculator, and for the forces involved the force calculator. The ratio calculator handles ratio arithmetic in general, the electrical power calculator covers the motor side of a drivetrain, and the speed converter turns wheel speed into road speed. The full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
Divide the number of teeth on the driven gear by the number on the driver. For a train of several meshes, multiply the individual stage ratios together. Pitch diameters can be used instead of tooth counts and give exactly the same result.
No. Power is torque multiplied by angular speed, and a gear train multiplies one while dividing the other by the same factor. In the ideal case power is unchanged, and in reality it falls slightly because friction turns some of it into heat.
Nothing. Its tooth count appears once as a numerator and once as a denominator and cancels exactly. An idler exists to change the direction of rotation or to bridge a distance between shafts, not to change speed or torque.
Each external mesh reverses the direction, so an odd number of meshes gives an output turning opposite to the input and an even number restores the original sense. Internal gear pairs, where a pinion runs inside a ring, do not reverse direction.
Whichever you have. Meshing gears must share a tooth pitch, so pitch diameter is directly proportional to tooth count and the two ratios are identical. Use pitch diameter rather than outside diameter, since the teeth mesh on the pitch circle.
It reduces the output torque and leaves the output speed unchanged, because speed is fixed rigidly by the tooth counts while losses appear as heat. Efficiencies compound across stages, so three stages at 97 per cent deliver about 91.3 per cent overall.
The torque converter rescales a torque figure between newton metres, pound-feet and other units. This calculator derives an output torque from an input torque and a gear ratio, producing a value that was never entered. One reformats a quantity, the other computes one.
This tool is provided for educational and estimating use. It models ideal gear kinematics with a flat efficiency factor and does not account for tooth strength, backlash, thermal limits or dynamic loading, so treat its output as a physics result rather than a gearbox design.
