How To Calculate Gear Ratio

A gear ratio is the number of teeth on the driven gear divided by the number on the driving gear, and it describes how one shaft's rotation is converted into another's. A 36 tooth gear driven by a 12 tooth gear gives a ratio of 3, meaning the input must turn three times for the output to turn once. The definition is a simple division, and everything else follows from it.

Tooth counts rather than diameters are used because two meshing gears must have teeth of the same pitch to work at all. Matching pitch means the tooth count is proportional to the circumference, so the ratio of teeth equals the ratio of diameters and either can be used. Counting teeth is simply the more reliable measurement.

A ratio above one is called a reduction, and it slows the output while multiplying its torque. The 3 to 1 example above turns 3000 revolutions per minute at the input into 1000 at the output, and turns 40 newton metres of input torque into 120. That is the trade every gearbox makes: speed is exchanged for turning force, and the product of the two stays roughly constant.

A ratio below one is an overdrive, and it does the opposite. With 40 teeth on the driving gear and 20 on the driven gear the ratio is 0.5, so the output turns twice as fast as the input and carries half the torque. Overdrives exist to let an engine run slowly at cruising speed without losing road speed.

The conservation of power is what makes the trade exact. Power is torque multiplied by angular speed, so if the output turns a third as fast it must carry three times the torque for the power to balance. Real gearboxes lose a few percent to friction, which is why the product is slightly lower on the output side, but the principle sets the ratio.

Direction reverses when an odd number of gears sits between input and output. Two meshing gears turn in opposite directions, and adding a third reverses the reversal, restoring the original sense. Idler gears used only to change direction do not alter the ratio at all, because their tooth counts cancel between the two meshes.

Multiple stages multiply together. A 3 to 1 stage feeding a 2 to 1 stage produces an overall ratio of 6, and the final output speed is the input speed divided by 6. This is how a gearbox reaches large ratios without any single pair of gears becoming impractically large.

Ratios are usually written with a colon, as in 3 to 1, and it is worth being clear which number refers to which shaft. A 3 to 1 reduction and a 1 to 3 overdrive are different arrangements, and reversing them turns a torque multiplier into a speed multiplier. The ratio itself is a single number, and the direction of the change is read from whether it exceeds one.

The relationship to speed is a division and to torque a multiplication, which is a common place to slip. Output speed equals input speed divided by the ratio, and output torque equals input torque multiplied by the ratio. In a reduction both results follow from the same number in opposite directions.

Real gears also have a pitch radius that sets the centre distance between shafts, and that distance matters when designing a housing. For the ratio itself, however, only the tooth counts enter. A designer picks the ratio first from the speed and torque requirement, then chooses tooth counts and module to realise it.