What a gear ratio is and what it buys you
A gear ratio is a tooth-count fraction: the driven gear's teeth divided by the driving gear's teeth. A 12-tooth pinion driving a 48-tooth gear is 48 ÷ 12 = 4, written 4:1. The output shaft turns a quarter as fast and carries close to four times the torque.
The reason it must be tooth counts rather than diameters is that meshing gears are obliged to share a module (metric) or diametral pitch (inch), which is the ratio of pitch diameter to tooth count. Sharing that constant means pitch diameter is exactly proportional to tooth count, so the tooth ratio is the pitch-diameter ratio with no measurement error attached. You can count teeth on an assembled gearbox; you cannot easily measure a pitch diameter on one.
What a ratio cannot do is create power. Power is torque multiplied by angular velocity, and a gear mesh conserves it apart from friction, so whatever you gain in torque you pay for in speed. A stage at 97% efficiency returns 3.88 times the input torque on a 4:1 reduction, not 4.00 times, and the missing 3% has become heat in the tooth flanks and the bearings.
Stages multiply, and so do their losses
A compound train puts two gears on a common shaft so that the second stage's driving gear turns at the first stage's output speed. The ratios then multiply: 4:1 followed by 3:1 is 12:1 overall. Three stages of 4, 3 and 2 give 24:1. That multiplication is the reason gearboxes are built in stages at all — a single 24:1 spur stage would need a driven gear twenty-four times the pinion's diameter, which is a very large casting to house.
Efficiency multiplies too, and in the unfavourable direction. At 97% per mesh, one stage returns 97%, two return 0.97² = 94.09%, three return 0.97³ = 91.27%. The loss per stage stays constant but the compounding is what makes a five-stage reduction noticeably warm. It is also why worm gearing, which can fall below 50% per mesh, is almost never compounded.
Direction alternates. Every external mesh — two spur or helical gears whose teeth face each other — reverses the sense of rotation, so an odd number of external meshes gives an output turning opposite to the input and an even number gives the same direction. Two exceptions matter in practice: an internal ring gear meshing with a pinion does not reverse direction, and an idler gear inserted purely to bridge a centre distance adds a mesh (changing the direction) without changing the ratio at all, because its tooth count appears once as driven and once as driving and cancels.
Worked example: a two-stage reduction from a 1,750 rpm motor
A four-pole induction motor runs at 1,750 rpm and delivers 10 N·m. It drives a 12-tooth pinion meshing with a 48-tooth gear; on that gear's shaft is a 15-tooth pinion meshing with a 45-tooth output gear. Assume 97% efficiency per mesh.
- Stage 1 ratio. 48 ÷ 12 = 4.000.
- Stage 2 ratio. 45 ÷ 15 = 3.000.
- Overall ratio. 4.000 × 3.000 = 12.000:1.
- Output speed. 1,750 ÷ 12 = 145.8 rpm.
- Overall efficiency. 0.97² = 0.9409, or 94.09%.
- Output torque. 10 × 12 × 0.9409 = 112.9 N·m, which is 83.3 lb·ft.
- Power check. Input power is 10 × 2π × 1,750 ÷ 60 = 1,833 W. Output power is 112.9 × 2π × 145.8 ÷ 60 = 1,724 W, and 1,724 ÷ 1,833 = 0.941 — the same 94.09%, which is the arithmetic check that the two calculations agree.
Direction: two external meshes, an even number, so the output turns the same way as the motor. Add a single idler anywhere in the train and it reverses without the ratio changing.
Choosing tooth counts, not just a ratio
The ratio is only part of the specification. Three other constraints usually decide the actual tooth counts.
Minimum pinion size. Standard 20° pressure angle full-depth involute teeth undercut at the root when the pinion falls below about 17 teeth generated with a rack-form cutter, which weakens the tooth exactly where it is most stressed. Below that, you need profile shift, a 25° pressure angle, or a stub tooth form. Very small pinions do exist — they are just not standard geometry.
Hunting tooth counts. If the two tooth counts share a common factor, the same pairs of teeth mesh repeatedly and any manufacturing error concentrates on those pairs. Making the counts relatively prime — 13 and 51 rather than 12 and 48 — means every tooth eventually meets every other tooth, which distributes wear and is standard practice on gears expected to run for years.
Ratio per stage. A single spur or helical stage above roughly 6 or 7 to 1 forces an awkwardly large driven gear. Splitting the reduction usually gives a smaller, cheaper, quieter box even after the extra mesh loss, which is why an industrial 25:1 gearmotor is almost always 5 × 5 rather than a single stage.
Finally, sanity-check the output torque against what the driven machine can survive. A gear train multiplies torque including shock torque, so a jam at the output can back-drive far more torque into the input than the motor's rating suggests. That is what shear pins, torque limiters and slip clutches exist to catch.
Output speed and torque multiplication from a 1,750 rpm input
| Arrangement | Stages | Overall ratio | Output speed (rpm) | Overall efficiency | Torque multiplier |
|---|---|---|---|---|---|
| Single 4:1 | 1 | 4.000 | 437.5 | 97.00% | 3.880 |
| Single 6:1 | 1 | 6.000 | 291.7 | 97.00% | 5.820 |
| 4:1 then 3:1 | 2 | 12.000 | 145.8 | 94.09% | 11.291 |
| 5:1 then 5:1 | 2 | 25.000 | 70.0 | 94.09% | 23.523 |
| 4:1, 3:1, 2:1 | 3 | 24.000 | 72.9 | 91.27% | 21.904 |
| 5:1, 4:1, 3:1 | 3 | 60.000 | 29.2 | 91.27% | 54.760 |
| Overdrive 1:2 | 1 | 0.500 | 3500.0 | 97.00% | 0.485 |
Compare rows three and five: 24:1 in three stages returns a lower torque multiplier relative to its ratio than 12:1 in two, because the third mesh costs another 3%.
Assumptions and things this does not cover
- Efficiency is treated as a constant per mesh. Real efficiency varies with load, speed, lubricant viscosity and tooth geometry, and it falls sharply at very light load where churning and bearing drag dominate.
- Bearing and seal losses are not separated out. The per-mesh figure you enter should be a whole-stage figure if you want a realistic answer for a complete gearbox.
- Idler gears do not change the ratio. An idler's tooth count appears as both driven and driving and cancels exactly. It changes direction and centre distance only, so do not enter it as a stage.
- Planetary and worm trains follow different rules. A planetary ratio depends on which member is held; a worm ratio is teeth divided by starts, and worm efficiency is strongly dependent on lead angle.
- Backlash and torsional wind-up are ignored. Both matter for positioning accuracy, and backlash referred to the output is divided by the ratio while input-side compliance is amplified by it.
- Nothing here says the teeth are strong enough. Bending and contact stress capacity are separate AGMA calculations that depend on face width, material, quality grade and the application factor.
Gears alongside belts, chains and the shafts they mount on
A gear train is one of three common ways to change speed, and the choice is usually about centre distance and stiffness rather than ratio. Gears are compact, stiff and efficient but require precise centre distances and make noise. Belts tolerate misalignment, absorb shock and cost far less, at the price of slip on friction drives and a much larger footprint — the pulley speed and ratio calculator solves the equivalent speed relationship for a belt drive, and the V-belt length calculator sizes the belt once the pulleys are chosen. Chains sit between the two: positive engagement like gears, long centre distances like belts.
The arithmetic transfers directly. In a belt drive the pitch diameters take the place of tooth counts, so N₁D₁ = N₂D₂ is the same statement as the tooth-count ratio; in a toothed timing belt the tooth counts are used directly and the drive is as positive as a gear train.
Mounting a gear to a shaft is its own decision, and it is usually a fit rather than a fastener: an H7/s6 shrink fit can transmit substantial torque by friction alone, which the ISO hole and shaft fit calculator quantifies, while a bolted hub relies on clamp load that the bolt torque calculator works out. AGMA standards govern the tooth geometry, quality grades and rating methods that decide whether the gears you have chosen will actually survive the torque this calculator predicts.