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Gear ratio is the number of teeth on the driven (output) gear divided by the number of teeth on the driving (input) gear. A 12-tooth pinion turning a 36-tooth gear gives 36 ÷ 12 = 3 : 1: the output turns at one third of the input speed and, apart from friction losses, delivers three times the torque.
Enter the two tooth counts and, optionally, the input speed. The calculator returns the ratio, the output RPM (input RPM ÷ ratio) and the ideal torque multiplier. A ratio above 1 is a reduction (slower, stronger output); below 1 it is an overdrive (faster, weaker output).
The definition is simple, but conventions differ between fields, and that is where most confusion comes from. Car people quote axle ratios such as 3.73 that match this calculator's driven-over-driving definition, while cyclists usually quote chainring over cog, which is the inverse. The sections below cover multi-stage gear trains and idler gears, the ratio range and efficiency of each gear type from NASA's gearing reference, how an axle ratio turns engine RPM into road speed, and how to read bicycle gearing with the same tool.
Gear Ratio Calculator: step-by-step worked example
- Driving gear teeth
- 14
- Driven gear teeth
- 38
- Input speed (rpm)
- 1,500
- 1
Gear ratio
Ratio = Driven teeth ÷ Driving teeth
38 ÷ 14 = 2.7143
- 2
Ratio notation
Ratio : 1 (> 1 reduction, < 1 overdrive)
= 2.7143 : 1
- 3
Output speed
Output rpm = Input rpm ÷ Ratio
1,500 ÷ 2.7143 = 552.6316 rpm
- 4
Torque multiplier
Output torque ≈ Input torque × Ratio (lossless)
= 2.7143 ×
- Input (driving)1,500 rpm
- Output (driven)552.6316 rpm
Good to know: Speed and torque trade off: a 2.71 : 1 ratio turns 2.71× slower but gives 2.71× the torque (minus friction losses). Idler gears in between change direction, not the ratio.
Gear Ratio Calculator — how it works
Ratio, speed and torque
| Quantity | Formula | Example: 12 driving, 36 driven, 1,800 rpm in |
|---|---|---|
| Gear ratio | driven teeth ÷ driving teeth | 36 ÷ 12 = 3 |
| Output speed | input rpm ÷ ratio | 1,800 ÷ 3 = 600 rpm |
| Output torque (ideal) | input torque × ratio | 10 N·m × 3 = 30 N·m |
| Output torque (real) | input torque × ratio × efficiency | 30 × 0.98 ≈ 29.4 N·m |
Speed and torque trade against each other because power (torque × angular speed) cannot increase through a gearbox; it can only be lost to friction. The calculator's torque multiplier is the ideal value, with no losses. For an efficient spur pair that is close to reality; for a worm drive it can overstate the output considerably, as the efficiency table below shows.
Tooth count works as the measure of ratio because meshing gears must have the same tooth spacing (circular pitch, or module), so the number of teeth is proportional to the pitch diameter. NASA's gearing reference notes that for worm sets, where pitch diameters are not directly related to tooth counts, the ratio must be taken from the number of teeth: worm wheel teeth divided by the number of threads (starts) on the worm.
Gear trains, idlers and direction of rotation
When several pairs are in series, the overall ratio is the product of the individual ratios. MIT's gear train lesson uses a hobby servo whose four stages multiply to a reduction of several hundred to one. Two stages of 3 : 1 and 4 : 1 give 12 : 1.
| Arrangement | Overall ratio | Output direction |
|---|---|---|
| One external pair | driven ÷ driving | opposite to input |
| Pair with one idler between | same as without the idler | same as input |
| Pair with two idlers | same as without the idlers | opposite to input |
| Compound train (two gears on one shaft) | product of each stage | depends on number of meshes |
An idler gear sits between the driver and the driven gear. University lecture notes on gear trains point out that in a simple train the speed ratio is independent of the size and number of intermediate gears; idlers are used to span a larger center distance or to set the direction of the output. Each external mesh reverses rotation, so an odd number of meshes reverses the output and an even number keeps the input direction. This calculator handles one pair; for a compound train, run each stage and multiply the results.
Ratio range and efficiency by gear type
NASA Reference Publication 1152, Gearing, tabulates the practical ratio range and nominal efficiency of single-mesh gear types:
| Gear type | Shafts | Ratio range per mesh | Nominal efficiency |
|---|---|---|---|
| Spur | parallel | 1–5 | 97–99.5% |
| Helical | parallel | 1–10 | 97–99.5% |
| Double helical (herringbone) | parallel | 1–20 | 97–99.5% |
| Straight or spiral bevel | intersecting | 1–8 | 97–99.5% |
| Worm | nonintersecting | 3–100 | 50–90% |
| Double-enveloping worm | nonintersecting | 3–100 | 50–98% |
| Hypoid | nonintersecting | 1–50 | 50–98% |
Two practical consequences follow. A single spur pair is rarely used beyond about 5 : 1, so large reductions are built from several stages. And a 40 : 1 worm drive at 60% efficiency delivers about 24 times the input torque, not the 40 that the ideal multiplier shows.
Axle ratios: how "3.73 gears" become road speed
In a rear-wheel-drive axle, the pinion on the driveshaft drives the ring gear. An axle sold as "3.73" typically has 41 ring teeth and 11 pinion teeth: 41 ÷ 11 = 3.7273, which is exactly what this calculator returns with 11 driving and 41 driven. Numerically higher ratios (4.10, 4.56) are "lower" gears: more torque at the wheels and quicker acceleration, but higher engine RPM at a given speed.
Vehicle speed follows from the wheel RPM and the tire's circumference:
mph = engine rpm × tire diameter (in) ÷ (transmission ratio × axle ratio × 336)
The constant 336 is 63,360 inches per mile ÷ (60 minutes × π). In a 1 : 1 transmission gear, 3,000 rpm through a 3.73 axle on 31-inch tires is 3,000 × 31 ÷ (1 × 3.7273 × 336.1) ≈ 74 mph.
Bicycles use the opposite convention
Cyclists describe gearing as chainring teeth ÷ rear cog teeth, so a 50-tooth chainring with a 25-tooth cog is "2 : 1" and the rear wheel turns twice per pedal revolution. In this calculator the chainring is the driving gear and the cog is the driven gear, so the same setup shows 0.5 : 1, an overdrive, with 90 rpm cadence giving 180 rpm at the wheel. Both describe the same mechanism; the cyclists' number is simply the inverse of the engineering ratio.
Cyclists also use gear inches: (chainring ÷ cog) × wheel diameter in inches. A 50 × 25 combination on a 27-inch wheel is 2 × 27 = 54 gear inches, which makes gearing comparable across wheel sizes.
Choosing tooth counts
- Check the mesh limits first. Gears only mesh with the same pitch or module, so pick a pitch and then choose tooth counts.
- Split large reductions. A 20 : 1 reduction is easier as 4 : 1 × 5 : 1 than as one spur pair.
- Remember speed is divided, torque multiplied. If an electric motor runs at 3,000 rpm and the mechanism needs 150 rpm, the required overall ratio is 20 : 1.
Worked example
You are deciding between 3.73 and 4.10 rear axle gears for a truck with 31-inch tires, and want to know the wheel speed and road speed at 3,000 rpm in a 1 : 1 transmission gear.
For 3.73, enter 11 driving teeth (pinion) and 41 driven teeth (ring gear) with an input of 3,000 rpm. The calculator returns a ratio of 3.7273 : 1, an output of 804.88 rpm and a torque multiplier of 3.7273. One turn of a 31-inch tire covers π × 31 = 97.39 inches, so the truck travels 804.88 × 97.39 × 60 ÷ 63,360 = 74.2 mph.
For 4.10, enter 10 driving and 41 driven teeth: the ratio is 4.1 : 1 and the wheels turn at 731.71 rpm, which is 67.5 mph at the same engine speed. The 4.10 gears multiply torque at the wheels by 4.1 ÷ 3.7273 = 1.10, about 10% more than the 3.73s, at the cost of about 10% more engine RPM at any given highway speed.
Real torque at the wheels is somewhat lower than these ideal multipliers because of losses in the transmission and axle.
Frequently asked questions
How do you calculate gear ratio?+
Divide the number of teeth on the driven (output) gear by the number on the driving (input) gear. 36 driven teeth and 12 driving teeth give 36 ÷ 12 = 3 : 1.
What does a 3.73 gear ratio mean?+
The driveshaft turns 3.73 times for each turn of the wheels. It usually comes from a 41-tooth ring gear and an 11-tooth pinion: 41 ÷ 11 = 3.7273.
Is a higher gear ratio faster?+
No, a numerically higher ratio gives more torque and less speed for the same input RPM. A 4.10 axle accelerates harder than a 3.73 but turns the wheels about 9% slower at the same engine speed.
How do I calculate output RPM from a gear ratio?+
Divide the input RPM by the ratio. 1,500 rpm into a 3 : 1 reduction gives 500 rpm; into a 0.5 : 1 overdrive it gives 3,000 rpm.
How does gear ratio affect torque?+
Output torque is input torque × ratio × efficiency. A 3 : 1 spur reduction at 98% efficiency turns 10 N·m into about 29.4 N·m; the calculator shows the ideal multiplier without losses.
Do idler gears change the gear ratio?+
No. In a simple train the overall ratio depends only on the first driving gear and the last driven gear. An idler changes the direction of rotation and the distance between shafts, not the ratio.
How do I calculate the ratio of a compound gear train?+
Multiply the ratio of each stage. A 3 : 1 stage followed by a 4 : 1 stage gives 12 : 1 overall, and output speed is the input divided by 12.
What is the gear ratio of a 50/25 bike gear?+
Cyclists call it 2 : 1, because the wheel turns twice per pedal stroke. Entered here as 50 driving and 25 driven teeth, it shows 0.5 : 1, the same gearing expressed as an engineering ratio.
How efficient are gears?+
NASA's gearing reference lists nominal efficiencies of 97–99.5% for spur, helical and bevel gears and 50–90% for worm gears. Worm drives therefore deliver noticeably less torque than their ratio suggests.
Sources
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- Publisher:
- Kalkoro — Teknolojig Bilişim Teknolojileri A.Ş., İstanbul
- Written by:
- Ahmet Tur — Editor (Math, date & time, unit conversion, physics & chemistry, everyday tools)
- Reviewed by:
- Kübra Yıldız — Editor
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