Gear Ratio Calculator

What a gear ratio measures

A gear ratio compares how fast the input shaft turns with how fast the output shaft turns, and tooth counts give it to you without any measuring. Count the teeth on the driving gear, count the teeth on the gear being driven, then divide: ratio = driven teeth ÷ driver teeth. A 12 tooth driver and a 36 tooth driven gear come out at 36 ÷ 12 = 3.00, which is written 3.00 to 1.

That single number carries three facts. The output turns 3.00 times slower than the input. Torque goes up by the same 3.00, because the teeth act as a row of levers. And in a simple pair of spur gears the output turns the opposite way, since the teeth push on each other from opposite sides.

The same number comes out of the pitch diameters, because meshing gears share one tooth spacing, so the diameter ratio matches the tooth ratio. Measuring diameters is the weaker path, though: a caliper across the outside of a gear picks up the tooth tips, not the pitch circle, and reads roughly two tooth heights too large. Count teeth instead. Teeth are countable, and the count is stamped into most catalog gears.

Reduction, overdrive and which number comes first

Notation trips people up more than the math does. A ratio above 1 means reduction: 3.00 to 1 says the input turns 3.00 times for each turn of the output. A ratio below 1 means overdrive: 0.50 to 1 says the output spins twice as fast as the input. Both describe the same pair; the number moves depending on which side you call the input. This calculator always reports driven teeth ÷ driver teeth, so anything above 1.00 slows the output down and multiplies torque.

Torque and speed swap in exact proportion, minus a little friction. A 3.00 to 1 pair turns 1,800 rpm into 600 rpm and turns 20 lb-ft into 60 lb-ft at the output shaft. The table below keeps the driver at 12 teeth and walks the driven gear up in size.

One pair can only do so much. Ratios past about 5 to 1 with spur gears force a tiny pinion, and pinions below roughly 17 teeth on a standard profile get undercut and run rough and weak. Past that point the fix is to stack stages.

Compound trains multiply the stages

When one pair cannot deliver the ratio, put two pairs in series and multiply them. A 20 tooth driver into a 40 tooth gear is 40 ÷ 20 = 2.00. A 15 tooth driver into a 45 tooth gear is 45 ÷ 15 = 3.00. Linked on shared shafts, the train gives 2.00 x 3.00 = 6.00 overall, so 3,000 rpm in comes out at 500 rpm.

Losses stack the other way. A pair of spur gears runs near 97 percent efficient, so two stages keep 0.97 x 0.97 = 0.94 of the power. Speed still divides by exactly 6.00, but the torque multiplication lands near 6.00 x 0.94 = 5.64 instead of a clean 6.00. Size bearings and couplings for the torque figure, and expect the housing to warm up by the difference.

The middle shaft carries the full torque of stage 1 while turning at the stage 1 output speed, so in a homemade gearbox it is usually the shaft that fails first. Check it before the housing goes together.

Speed, torque and the power that stays put

Gears trade speed for torque, and the product of the two barely moves. That product is power. In US units, horsepower = torque in lb-ft x rpm ÷ 5,252, where 5,252 comes from 33,000 ft-lb per minute divided by 2π. A shaft carrying 20 lb-ft at 1,800 rpm makes 20 x 1,800 ÷ 5,252 = 6.85 hp. After the 3.00 to 1 reduction it turns 600 rpm with 60 lb-ft, and 60 x 600 ÷ 5,252 = 6.85 hp again. Nothing was created except mechanical advantage.

This is the fastest paper check there is for a gear train. Multiply torque by rpm at the input, do it again at the output, and the two answers should match within the friction loss. A big gap means a tooth count is wrong, a stage got counted twice, or the numbers came from different load points.

The same identity explains why torque and horsepower curves on an engine dyno always cross at 5,252 rpm: below it torque reads higher than horsepower, above it lower. Gears do not move that crossing. They move the operating point onto a better part of the curve, which is the entire reason a gearbox exists.

Gear size: module, pitch diameter and center distance

Tooth counts set the ratio, but the gears also have to fit the housing. Metric gears are sized by module, the pitch diameter in millimeters per tooth: module = pitch diameter ÷ teeth, so pitch diameter = module x teeth. A module 3 gear with 20 teeth has a pitch diameter of 3 x 20 = 60 mm, and a mating 60 tooth gear has 3 x 60 = 180 mm.

Center distance is half the sum of the two pitch diameters: (60 + 180) ÷ 2 = 120 mm. That is the dimension a bearing bore and a housing drawing care about. Working backward from a known center distance, module = 2 x center distance ÷ (teeth 1 + teeth 2); for 120 mm and 80 total teeth that gives 2 x 120 ÷ 80 = 3.

Inch gears flip the definition. Diametral pitch divides teeth by pitch diameter in inches: DP = teeth ÷ pitch diameter. A 32 tooth gear at DP 8 has a pitch diameter of 32 ÷ 8 = 4.00 in, a 64 tooth mate at the same DP has 64 ÷ 8 = 8.00 in, and the shafts sit (4.00 + 8.00) ÷ 2 = 6.00 in apart. The two systems are reciprocals, DP = 25.4 ÷ module, so module 3 works out near DP 8.47. A module gear and a DP gear will not mesh with each other, because the tooth spacing differs; keep one system per mesh.

Ratios you can check against real machines

A bicycle makes the idea visible. A 50 tooth chainring driving a 25 tooth rear cog is 50 ÷ 25 = 2.00, so at an 80 rpm pedaling rate the wheel turns 160 rpm. A 26 in wheel rolls 26 x 3.1416 = 81.68 in per turn, and 160 x 81.68 = 13,069 in per minute, which is 12.4 mph on the road. Cyclists compress the same arithmetic into gear inches: 26 x 50 ÷ 25 = 52 gear inches, the diameter of an imaginary direct drive wheel.

A car stacks two more ratios in series. A first gear of 3.50 to 1 and a final drive of 3.90 to 1 multiply to 13.65, so at 2,500 rpm the wheels turn 2,500 ÷ 13.65 = 183 rpm. With a 26.7 in tire that rolls 3.1416 x 26.7 = 83.88 in per turn, about 15,366 in per minute, which is 14.6 mph. That is why first gear feels short: it exists to move the car from rest, not to cruise.

Power tools hide reductions too. A 1,500 rpm motor behind a 4.00 to 1 gear set turns the chuck at 1,500 ÷ 4 = 375 rpm and multiplies torque by four, which is how a small motor drives a half inch bit without stalling. The calculator above handles all three shapes of problem: a single pair, a stacked train, and gear size with center distance.

Checks before you trust the number

Tooth counts beat every other method, but they have their own traps. Count twice, and count teeth at the pitch line rather than by spinning the gear while half watching. One missed tooth turns a 3.00 into a 2.94, and nobody notices that on a datasheet.

• Direction: a simple pair reverses rotation. An idler gear between them reverses it back and changes nothing about the ratio; idlers exist for spacing and direction, not reduction.
• Shared factors: two gears whose tooth counts share a factor, such as 17 and 51, bring the same teeth back together on every turn. Counts with no common factor spread the wear evenly, which gear builders call a hunting tooth layout.
• Efficiency: plan on 97 percent per spur mesh, and less for worm drives, where a steep lead angle can leave the output below 50 percent of what the ratio promises.
• Percentage thinking: a 3.00 to 1 reduction cuts output speed by 66.7 percent, not by 300 percent. The ratio counts input turns, not the size of the cut.
• Real road speed: tire diameter, belt slip and torque converter slip all sit outside the gear math. The gear inches convention folds the wheel in so the number compares across bikes.

For the vocabulary around tooth shapes and trains, the gear train overview and Britannica's gear article cover the terms without standards pricing.