How to calculate gear ratio

A gear ratio tells you how a pair of meshing gears trades speed for torque. It is one of the most useful numbers in all of mechanical design: from it you can predict the output shaft speed, the output torque, and — in a vehicle — the road speed for any engine RPM. The arithmetic is simple, but the sign conventions trip people up, so it is worth getting the definition exactly right before you reach for a calculator.

The gear ratio formula

The gear ratio is the number of teeth on the driven (output) gear divided by the number of teeth on the driving (input) gear:

i = z₂ / z₁ = driven teeth / driving teeth

Here z₁ is the input gear (often the pinion or the front sprocket) and z₂ is the output gear (the ring gear or rear sprocket). The result is usually written as a ratio against 1 — a ratio of 3 is spoken as “three to one” and written 3:1. The direction of the division matters: it is always output over input, never the reverse.

• i > 1 — reduction. The output turns slower than the input, and torque is multiplied. This is the common case: gearboxes, axle final drives and machine reducers almost always reduce.
• i < 1 — overdrive. The output turns faster than the input, and torque is divided. Think of a bicycle in its highest gear, or an overdrive top gear in a car.
• i = 1 — direct drive. Equal tooth counts; speed and torque pass straight through unchanged.

Speed and torque

Once you have the ratio, the output speed and torque follow immediately. Gears conserve power (less the small friction loss at each mesh), so whatever you gain in torque you pay for in speed:

n₂ = n₁ / i = n₁ · z₁ / z₂  (output speed)

T₂ = T₁ · i  (ideal output torque)

A 3:1 reduction therefore cuts the speed to one third and multiplies torque by three. The torque figure is the ideal value: a real spur or helical mesh is roughly 96–99% efficient, so each stage shaves off a few percent. Worm drives lose far more. For a quick, exact figure you can plug the numbers into the gear ratio calculator, which reports the ratio, output RPM, output torque and — if you give it a tire diameter — the road speed.

Worked example — a single reduction

Suppose a 12-tooth pinion drives a 36-tooth gear, with the pinion turning at 1800 RPM under 50 N·m of torque:

i = z₂ / z₁ = 36 / 12 = 3  (a 3:1 reduction)

n₂ = 1800 / 3 = 600 RPM

T₂ = 50 × 3 = 150 N·m  (ideal)

So the output shaft turns at 600 RPM with three times the torque — exactly the trade a reduction is for. To see only the tooth side of this calculation, the gear tooth ratio calculator takes the two counts and returns the ratio and output speed directly.

Multi-stage and compound gear trains

Most real gearboxes need more reduction than a single sensible pair can give, so they use several stages in series. The rule is simple: stage ratios multiply. For a compound train where stage 1 is z₂/z₁ and stage 2 is z₄/z₃:

i_total = (z₂ / z₁) · (z₄ / z₃) · …

A “compound” train is one where the driven gear of stage 1 and the driving gear of stage 2 are keyed to the same intermediate shaft, so they turn together. That shared shaft is what lets the reductions stack.

Two-stage example. Stage 1 is a 15T gear driving a 45T gear (3:1). On the same lay shaft a 16T gear drives a 64T gear (4:1):

i_total = (45 / 15) · (64 / 16) = 3 × 4 = 12  (12:1)

An input of 1800 RPM therefore leaves at 1800 / 12 = 150 RPM, and the input torque is multiplied by 12. Splitting a large reduction across two modest stages keeps each gear pair a sensible size and lets you hit ratios that a single pair never could.

Idler gears change direction, not ratio

An idler is a gear placed between the driving and driven gears, meshing with both. People often assume it alters the ratio — it does not. Work it through: with an idler of z_i teeth between input z₁ and output z₂, the two stage ratios are z_i/z₁ and z₂/z_i, and multiplying them cancels the idler entirely:

i = (z_i / z₁) · (z₂ / z_i) = z₂ / z₁

The only thing an idler does is reverse the direction of rotation. Each mesh flips the spin, so an idler is how engineers make the output turn the same way as the input, or fit a gear train into an awkward centre distance, without touching the ratio.

Gear ratio in vehicles

For cars, motorcycles and go-karts, “gear ratio” usually means the whole driveline. The engine speed reaches the wheels through the selected transmission gear and then the final drive (the differential or the chain-and-sprocket ratio). These multiply just like any other stages:

overall = transmission ratio × final-drive ratio

To turn the resulting wheel speed into road speed, the tire diameter does the work. Each wheel revolution rolls the car forward by one tire circumference, so:

v = π · d_tire · n_wheel

with n_wheel the wheel speed in revolutions per unit time. (Mind the units: if d_tire is in metres and n_wheel in rev/min, this gives metres per minute — divide by 60 for m/s, or by 1000 then ×60 for km/h.)

Worked road-speed example. Take the 3:1 single reduction from earlier: 600 RPM at the output, driving a wheel with a 0.633 m (≈ 633 mm) tire:

v = π × 0.633 × 600 = 1193 m/min ≈ 19.9 m/s ≈ 71.6 km/h (44.5 mph)

This is why tire size and gearing are inseparable. Fitting a taller tire raises the speed per wheel revolution — gaining top speed but reducing torque at the contact patch — exactly as if you had chosen a numerically lower (“taller”) final-drive ratio. Racers and off-roaders trade these two against each other constantly.

Common gear ratios

A few reductions show up again and again. The output RPM column assumes a 1800 RPM input, and the torque multiplier is the ideal (loss-free) figure:

Teeth (driving → driven)	Ratio	Output RPM (in = 1800)	Torque ×
20 → 20	1:1	1800	1.0
20 → 40	2:1	900	2.0
12 → 36	3:1	600	3.0
11 → 44	4:1	450	4.0
10 → 50	5:1	360	5.0
40 → 20	0.5:1 (overdrive)	3600	0.5

Common mistakes

• Counting teeth backwards. The ratio is driven ÷ driving — output over input. Flip it and a 3:1 reduction reads as a 0.33 overdrive, and your speed and torque predictions invert. When in doubt, sanity-check the sign: a gearbox that is meant to reduce should give i > 1.
• Forgetting that idlers don’t count. Only the first and last gear in a simple chain set the ratio; the gears in between (unless they are compound, sharing a shaft with the next pair) are idlers and cancel out. Including their teeth in the math is a classic error.
• Confusing ratio with module. The ratio comes from tooth counts; the module (or diametral pitch) is the tooth size and sets the physical gear diameter and centre distance. Two gears must share a module to mesh, but the module never appears in the ratio. If you need the geometry, use the involute and module tools rather than the ratio formula.
• Adding stage ratios instead of multiplying. A 3:1 stage and a 4:1 stage make a 12:1 train, not a 7:1 one. Reductions compound multiplicatively.

Beyond simple gear trains

Planetary (epicyclic) gear sets — used in automatic transmissions, hub motors and many compact reducers — follow different rules, because the ratio depends on which member (sun, ring or carrier) is held fixed and which is driven. For those, the simple driven-over-driving formula does not apply directly; the planetary gear calculator handles each fixed-member arrangement for you. For ordinary fixed-axis spur, helical and sprocket drives, though, i = z₂ / z₁ and its speed and torque consequences are all you need.

Frequently asked questions

What is the gear ratio formula?

Gear ratio is i = driven teeth ÷ driving teeth = z₂ / z₁ — the output gear over the input gear. A ratio above 1 is a reduction (the output turns slower but with more torque); below 1 is an overdrive (faster, less torque). Output speed is n₂ = n₁ / i and ideal output torque is T₂ = T₁ × i.

Do idler gears change the gear ratio?

No. An idler gear sitting between the driving and driven gears only reverses the direction of rotation — it does not change the overall ratio. The ratio still depends solely on the first and last gear in the chain, because the idler’s teeth cancel out (they appear in both the numerator and denominator when you multiply the stage ratios).

How do I find the ratio of a multi-stage gear train?

Multiply the individual stage ratios together: i_total = (z₂/z₁) × (z₄/z₃) × … Each meshing pair is one stage, and a compound train shares a shaft between stages so the speed reductions stack. For example a 3:1 stage feeding a 4:1 stage gives an overall 12:1 reduction.

How does gear ratio relate to vehicle speed?

The drivetrain ratio sets how fast the wheels turn for a given engine RPM, and the tire diameter converts that wheel speed into road speed: v = π × d_tire × wheel-rpm. A taller (larger) tire travels farther per revolution, so it raises top speed but reduces the effective torque at the contact patch — much like fitting a numerically lower final-drive ratio.

Last reviewed: 2026-05-29.