How this instrument works
Gear ratio counts teeth, and nothing else. Meshing wheels must share one tooth size, so teeth cross their contact line at an identical rate: a 20-tooth pinion spins three full turns to walk 60 teeth past that line, dragging its 60-tooth partner around exactly once. Divide driven teeth by driving teeth and you hold two facts at once — how many input turns buy one output turn, and, in an ideal train, how many times torque is multiplied on its way through.
Gearing long predates any theory of gearing. Bronze wheels inside the Antikythera mechanism, raised from a wreck in 1901 and dated to roughly 100 BCE, carry tooth counts chosen as astronomy rather than machinery — 223 teeth for one Saros eclipse cycle, 127 for lunar months spread across 19 years. Tooth shape arrived much later: Leonhard Euler argued in the 1760s for an involute profile, a curve holding angular speed steady through mesh instead of letting it surge tooth by tooth, and Robert Willis set out mechanism ratios systematically in 1841.
One caution: this sheet reports an ideal, rigid, single-stage answer. Real trains lose a little — roughly 1–2% per spur mesh, far more through a worm — so torque gain always falls short of ratio, and heat carries off that difference. Backlash, tooth deflection and chain stretch add small timing errors, which matter to machine tools and not to cement mixers. Ratios also stop being one division once geometry changes: compound trains multiply stage by stage, a planetary set answers differently depending on which member is held, and a CVT has no tooth count to divide.
- Enter Driven gear teeth — the wheel on your output shaft, the one being turned.
- Enter Driving gear teeth — the pinion on your motor or input shaft.
- Set Input speed in rpm, or switch to hertz if you think in revolutions per second.
- Read Gear ratio: above 1 is a reduction, below 1 an overdrive, exactly 1 a pass-through.
- Read Output speed, then multiply input torque by that ratio for ideal output torque.
Worked example — 3:1 under a 1500 rpm motor
A conveyor drive. Its four-pole motor turns 1500 rpm — 25 revolutions per second — and carries a 20-tooth pinion meshed to a 60-tooth wheel on a roller shaft. Gear ratio = 60 ⁄ 20 = 3, printed as 3:1. Output speed = 1500 × 20 ⁄ 60 = 500 rpm, which is 8.3333 revolutions per second in SI units.
That factor of three runs both ways. Roller speed drops to a third, while an ideal 12 N·m at that motor arrives as 36 N·m at your roller — nearer 35 N·m once mesh losses are paid. Swap pinion and wheel, and an identical pair overdrives instead: 500 rpm in, 1500 rpm out, torque cut to a third.
Questions
Which tooth count goes on top?
Driven teeth on top, driving teeth underneath. That convention makes a reduction come out above 1 — a 60-tooth wheel driven by a 20-tooth pinion is 3:1 — and lets you read ratio as an ideal torque multiplier straight off. Invert it and you get a speed multiplier instead, which is why a bicycle's 53/11 top gear gets quoted as 4.8 rather than 0.21. Both numbers are true; only one matches this sheet.
Does a gear ratio have a unit?
No. Teeth divided by teeth leaves a pure number, conventionally written with a colon: 3, 3:1 and 3.000 all say one thing. Because pitch is shared across any working mesh, pitch diameters carry an identical ratio to tooth counts, so measuring two wheels with calipers agrees with counting them — handy when a gear sits inside a housing you would rather not open.
Does an idler gear change my ratio?
No. An idler reverses rotation direction and does nothing else. Its teeth appear once above and once below in a full train ratio, cancelling exactly, so a 13-tooth idler and a 40-tooth idler between one pair produce identical output speed. Designers fit them for direction, for centre distance, or to keep two shafts turning together.
Does 3:1 really triple torque?
Almost. Power out equals power in minus losses, and since speed falls by three, torque must climb by nearly three. A well-cut, well-lubricated spur mesh returns roughly 98–99%, so 36 N·m ideal shows up as about 35 N·m. Worm gearing is a sharp exception: one worm stage can reach 60:1 yet hand back only 50–90% of input power, much of it lost as heat in sliding contact.
Why do designers avoid whole-number ratios?
In an exact 3:1 pair, three particular pinion teeth meet one particular wheel tooth forever, so a single hard particle or grinding flaw wears that spot repeatedly. Adding a hunting tooth — 61 rather than 60 — makes counts relatively prime, so every tooth eventually meets every other and wear spreads evenly. Ratio shifts to 3.05:1, a change most drives absorb without complaint.
Does this apply to chains, belts and planetary sets?
Chain and toothed-belt drives, yes: count sprocket or pulley teeth and use them directly, since neither slips. Flat belts and friction wheels need pitch diameters plus an allowance for creep. Planetary gearsets do not fit this form at all — output depends on which of sun, ring or carrier is held, so a sun-input, ring-fixed set gives 1 + N_ring ⁄ N_sun, not one division.