⚙️ Gear Ratio

Written and maintained by the PhDino author · Last reviewed 21 September 2026 · Checked against 2 independent reference calculations · how PhDino checks its numbers

How tooth count determines the speed and torque relationship between two meshing gears.

Two gears meshing together must have teeth that move past the mesh point at the same linear speed — which means the gear with more teeth turns proportionally slower. That speed relationship inverts for torque: for an ideal (lossless) gear pair, whatever speed ratio you gain, you trade for the reciprocal in torque. This is exactly analogous to a lever trading force for distance.

Real gear trains lose some of that torque to friction, so actual output torque is always somewhat less than the ideal ratio predicts — the efficiency loss depends on gear type, lubrication, and load.

Key formula

Gear ratio = N_driven / N_driving
Output speed = Input speed / Gear ratio
Output torque ≈ Input torque × Gear ratio (ideal, no losses)

Variables

N_driving, N_driven

tooth counts of the driving and driven gears

How to use the Gear Ratio calculator

Use this to see what a pair of meshing gears (or two sprockets, or two pulleys) does to speed and torque: how fast the output turns, and how much the torque is multiplied on the way. It answers the everyday design question of how to get a slow, strong output from a fast motor, or the reverse.

The convention here is driven teeth divided by driving teeth. A ratio above 1 is a reduction, so the output is slower and stronger than the input; a ratio below 1 is an overdrive. Some fields quote the ratio the other way round (cyclists give chainring over cog), so check which convention a spec uses before comparing numbers.

Driving Gear Teeth

The number of teeth on the input gear, the one connected to the motor or engine. This is the small pinion in a typical reduction.

Driven Gear Teeth

The number of teeth on the output gear, the one connected to the load. Use the true tooth count; the two gears must share the same module or pitch to mesh at all.

Driving Speed (RPM)

The rotational speed of the input shaft in RPM. For an electric motor use the loaded speed on the nameplate, which is a little below the synchronous speed.

Worked example: a motor and a single-stage reducer

A 1,750 RPM motor drives a hoist drum through one pair of spur gears. The pinion on the motor shaft has 15 teeth and it meshes with a 60-tooth gear on the drum shaft. What speed and torque does the drum see?

Worked by hand:

1. Ratio. Driven teeth ÷ driving teeth = 60 ÷ 15 = 4.00, so this is a 4-to-1 reduction.
2. Output speed. Speed falls by the ratio: 1,750 ÷ 4.00 = 437.5 RPM.
3. Torque. In an ideal mesh power is conserved, so torque rises by the same factor the speed falls: 4.00 times the input torque.
4. Allow for losses. A well-made spur mesh is around 97% efficient, so the usable multiplication is nearer 4.00 × 0.97 = 3.88.

The drum turns at 437.5 RPM with about 3.9 times the motor's torque. For a bigger reduction, chain stages: two stages of this ratio give 16 to 1 and an output of 109.4 RPM, because the ratios of stages multiply.

Reading the result: what a ratio really tells you

A gear pair trades speed for torque and cannot make power. Torque multiplication is exactly as large as the speed reduction, less friction, so a 4:1 reducer gives a fourth of the speed and just under four times the torque, never more. If the output needs both more speed and more torque than the motor supplies, the answer is a bigger motor, not a different ratio.

• Each external mesh reverses the direction of rotation. An idler gear between the two changes the direction back without changing the ratio, because only the first and last gears count.
• Tooth counts also set practical limits. Very small pinions (below about 17 teeth for standard 20° involute gears) undercut at the root and weaken, so large single-stage ratios are usually built as two or three stages.
• Choose tooth counts with no common factor when you can. A 15-tooth pinion on a 61-tooth gear brings each tooth against every other tooth in turn and spreads wear evenly, while 15 on 60 meets the same few teeth every fourth revolution.
• The same arithmetic holds for chains and toothed belts (count teeth) and for smooth belts (use pitch diameters in place of teeth).

Notes & limitations

• A ratio greater than 1 (more teeth on the driven gear) means a speed reduction and a torque increase — this is the far more common arrangement in machine design, since most motors run fast and most loads need torque more than speed.
• This formula assumes two gears meshing directly; a gear train with idler gears or multiple stages compounds the ratio stage by stage.

Common mistakes

• Swapping driving and driven. Doing so turns a reduction into an overdrive, and the calculator reports the reciprocal of what was intended.
• Expecting torque for free. Output torque multiplies by the ratio only because the speed drops by it; there is no way to gain both.
• Ignoring efficiency. Each stage loses a few percent to friction, and worm gears can lose far more, so a multi-stage train delivers noticeably less than the ideal figure.
• Forgetting the mating requirements. Two gears with the same tooth counts as another pair still will not run together unless their module or diametral pitch and pressure angle match.
• Picking a ratio that exceeds what the pinion can survive. The tooth loading on the small gear rises with torque, and the pinion is nearly always the limiting part.

Frequently asked questions