Gear Ratio Calculator

A gear ratio calculator turns two tooth counts into the number that governs every gearbox, axle and bicycle drivetrain: ratio = driven teeth ÷ driver teeth. A 20-tooth driver meshing with a 60-tooth driven gear is a 3:1 reduction, so 1,800 RPM in becomes 600 RPM out and 50 lb-ft of torque becomes 150 lb-ft. Chain up to three pairs, or switch to bicycle mode for gear inches and speed at your cadence.

This page is the mechanical cousin of a few other tools on the site. The Torque Calculator finds torque from force and lever arm, the Horsepower Calculator turns torque and RPM into power, and the Tire Size Calculator shows how a different tire diameter changes your effective final drive. Come here when the question is "what does this pair of gears do to speed and torque?"

The gear ratio formula

Two gears in mesh must move the same number of teeth past the contact point in the same time. That single fact produces every formula on this page:

The driver is the gear that receives power (from a motor, an engine or your pedals). The driven gear is the one that delivers it. When the driven gear is larger, the ratio is greater than 1 and you get a reduction: slower output, more torque. When the driven gear is smaller, the ratio is below 1 and you get an overdrive: faster output, less torque. Power is conserved in an ideal gear pair, which is why torque rises by exactly the factor that speed falls.

Worked example: a single 3:1 reduction

Suppose a small motor spins at 1,800 RPM and delivers 50 lb-ft of torque to a 20-tooth pinion. The pinion drives a 60-tooth gear. Step by step:

1. Ratio: 60 ÷ 20 = 3.0, written 3:1. The pinion turns three times per turn of the big gear.
2. Output speed: 1,800 ÷ 3 = 600 RPM.
3. Output torque: 50 × 3 = 150 lb-ft (ideal, before friction).
4. Speed change: the output turns at 1 ÷ 3 = 0.333 of the input, a 66.7% reduction.

Now let a 40-tooth gear drive a 20-tooth gear and you have an overdrive of 20 ÷ 40 = 0.5:1. Now 1,800 RPM in becomes 3,600 RPM out and 50 lb-ft shrinks to 25 lb-ft. Same hardware, opposite job.

Worked example: a three-stage gear train

Gearboxes rarely get a large reduction from one pair, because the driven gear would have to be enormous. Instead they stack stages. Take a 3,000 RPM motor producing 10 lb-ft, feeding three pairs of 12:36, 15:45 and 20:40 teeth:

The overall ratio is 3 × 3 × 2 = 18:1. Output speed is 3,000 ÷ 18 = 166.67 RPM and ideal output torque is 10 × 18 = 180 lb-ft. Notice that the order of the stages does not change the final answer, only the intermediate shaft speeds. In a real box each mesh loses a little power; at 97% per stage the train passes on 0.97³ = 91.3% of the input power, so the shaft would actually see closer to 164 lb-ft.

Common gear pairs at a glance

Every row assumes a 1,800 RPM input, a typical speed for a four-pole induction motor. The torque column is the multiplier applied to whatever input torque you have:

Three different pairs, 12:36, 15:45 and 20:60, all produce exactly 3:1. Larger tooth counts do not change the ratio; they spread the load over more teeth and run more smoothly, which is a strength question rather than a ratio question.

Bicycle gear ratio and gear inches

A bike drivetrain is a chain drive, so the front chainring is the driver and the rear cog is the driven sprocket. Cyclists flip the fraction and quote chainring ÷ cog, so a bigger number means a harder, faster gear. Because that ratio alone ignores wheel size, the traditional yardstick is gear inches:

A 50-tooth chainring and 17-tooth cog on a 27-inch road wheel give 27 × 50 ÷ 17 = 79.4 gear inches. Each full pedal turn moves the bike 79.4 × π = 249.5 inches, or 20.79 feet. At a cadence of 90 RPM that is 249.5 × 90 × 60 ÷ 63,360 = 21.26 mph (34.2 km/h). Drop to the 34-tooth small ring and a 28-tooth cog and you are in a 32.8-inch climbing gear that moves just 8.8 mph at the same cadence.

Gear inches table: road chainrings on a 27-inch wheel

The table pairs the four most common road chainrings with a spread of cassette cogs. Values are wheel diameter × chainring ÷ cog:

A compact 50/34 crankset with an 11-28 cassette therefore runs from 32.8 to 122.7 gear inches, a range of 3.74 to 1. Note the overlaps: 34 × 17 (54.0) and 50 × 25 (54.0) are the same gear reached two different ways, which is why cross-chaining rarely buys you anything.

Speed by gear inches and cadence

Because speed = gear inches × π × cadence × 60 ÷ 63,360, every 10 gear inches adds about 2.68 mph at 90 RPM. This table shows the result in mph:

How to use this gear ratio calculator

1. Pick a mode. Single pair for two gears, sprockets or pulleys; Multi-stage for a gear train of up to three pairs; Bicycle for chainring, cog and wheel size.
2. Enter tooth counts. Count the teeth on the driver (input) and driven (output) gear. For pulleys or friction wheels, enter diameters instead; the ratio math is identical.
3. Add input speed and torque if you want output values, not just the ratio. Speed is in RPM and torque in lb-ft; if your data is in newton-meters, the Unit Converter handles the conversion.
4. Read the headline ratio, then the output speed and ideal torque in the tiles below it. The table under the result shows the same gear pair at five other input speeds.
5. In bicycle mode, tap a wheel size (26", 27" for 700c road wheels, 27.5" or 29") or type an exact diameter, set your cadence, and scan the cassette table to see every gear's speed.

Who this calculator is for

• Hobby machinists and robotics builders sizing a gearmotor or 3D-printed gear train to hit a target output speed.
• Car and truck owners comparing axle ratios (3.73 versus 4.10, say) before a differential swap or a tire size change.
• Cyclists choosing a cassette or chainring, checking whether a single-speed ratio is right, or converting between gear inches and mph.
• Students working through mechanical-advantage problems in physics or engineering courses.
• RC, go-kart and e-bike tinkerers matching motor RPM to wheel speed through a pinion and spur gear.

Key terms

• Driver / driven: the input gear and the output gear of a pair. Swapping them inverts the ratio.
• Pinion: the smaller gear in a pair, usually the driver in a reduction.
• Reduction: a ratio above 1:1 that slows the output and multiplies torque. Most gearboxes are reductions.
• Overdrive: a ratio below 1:1 that speeds the output up and divides torque; the top gear of many transmissions.
• Idler gear: a gear placed between driver and driven. It reverses rotation but does not change the ratio, because it is both driven and driver with the same tooth count.
• Final drive / axle ratio: the last reduction before the wheels of a vehicle, typically 3.0:1 to 4.5:1.
• Gear inches: the bicycle gear yardstick, equal to the diameter of a directly driven wheel that would give the same distance per pedal turn.
• Development: the distance a bicycle travels per crank revolution, equal to gear inches × π.

Factors that change the result

The ratio itself depends only on tooth counts, but what the ratio does for you depends on everything downstream:

• Stacked stages multiply. Two 3:1 stages are 9:1, three are 27:1. Small changes to one stage scale the whole train.
• Tire or wheel diameter acts like one more gear stage. A 28-inch tire turns 720.3 times per mile; through a 3.73 axle and a 0.75 overdrive gear, the engine spins 720.3 × 3.73 × 0.75 = 2,015 RPM at 60 mph. Swap in a 4.10 axle and it climbs to 2,215 RPM, 9.9% higher.
• Efficiency trims torque but never speed. Speed is fixed by geometry; torque is what friction eats.
• Cadence on a bike scales speed directly: 79.4 gear inches is 14.2 mph at 60 RPM and 23.6 mph at 100 RPM.
• Direction of rotation flips with every external gear mesh but not with chains and belts. It does not affect the numbers, only which way the output turns.

Tips for choosing a ratio

• Work backwards from the output you need. If a wheel must turn 100 RPM from a 1,725 RPM motor, you need about 17.25:1, which splits neatly into two stages of roughly 4:1 and 4.3:1.
• Keep any single spur-gear stage under about 6:1 to 8:1; beyond that the driven gear gets large and the pinion small and weak. Add a stage instead.
• For bicycles, target the two extremes first: the hardest gear you can spin at 100 RPM on a descent and the easiest you can turn at 60 RPM on your steepest climb. Everything in between fills itself in.
• When comparing axle ratios, look at RPM at your cruising speed, not just 0-60 feel. Fuel economy tracks engine speed.
• Prime tooth counts (such as 17 or 43) wear more evenly because the same two teeth rarely meet, which is one reason odd numbers like 11:43 show up in axles.

Limitations

• All results are ideal. No friction, bearing drag or windage is deducted, so real torque is a few percent lower per stage and much lower for worm drives.
• The calculator handles simple and compound trains of external gears, chains and belts. Planetary gear sets, differentials and CVTs follow different formulas.
• It says nothing about strength: whether a 12-tooth plastic pinion survives 180 lb-ft is a materials question.
• Bicycle wheel sizes are nominal. A 700 × 25c tire is closer to 26.3 inches than 27, which turns the 50/17 gear into 77.4 gear inches. Enter a measured diameter for precision.
• Vehicle examples ignore tire slip, torque-converter multiplication and transmission losses.

Related calculators and when to use which

Use this page when the unknown is the ratio, output RPM or torque of a gear, chain or belt pair. Use the Torque Calculator when you start from a force and a lever arm rather than an existing torque figure, the Horsepower Calculator when you want to convert the torque and RPM you get here into power, the Tire Size Calculator when a tire change rather than a gear change is what alters your effective ratio, and the Speed Distance Time Calculator when you already know your speed and just need travel time.