Gear Ratio Calculator – The Output Speed and Torque Your Gears Really Give

Reduction, overdrive and direct drive, in one table

Every gear pair on this page does the same three things at once: it changes the speed, it changes the torque by the inverse factor, and, for a single mesh, it reverses the direction. The ratio is the whole story, and it is simply the driven teeth divided by the driver teeth. A ratio above 1 is a reduction: the output turns more slowly and twists harder. A ratio below 1 is an overdrive: the output turns faster and twists more weakly. A ratio of exactly 1 is direct drive: speed and torque pass through unchanged, and only the direction flips. The table below was produced by this page’s own calculator at an input speed of 3,000 RPM and an input torque of 10 N·m, so you can reproduce every row by typing the same teeth counts in.

Read the reduction rows first, because they show the trade in its cleanest form. The default pair, 12 driving 40, gives a ratio of 40 divided by 12, which is 3.33, and the output speed is 3,000 divided by 3.33, which is 900 RPM, while the output torque is 10 times 3.33, which is 33.33 N·m. The 20-to-40 pair halves the reduction: the ratio is 2.00, the output is 1,500 RPM and the torque doubles to 20.00 N·m. The 10-to-40 pair doubles it the other way: the ratio is 4.00, the output falls to 750 RPM and the torque rises to 40.00 N·m. In every reduction row the speed factor and the torque factor are reciprocals, and their product is the input power, apart from friction. Nothing is created: the extra torque at the output is paid for, exactly, by the lost speed.

The overdrive row is the same trade run backwards, and it is the row most people get wrong on first sight. With 40 teeth driving 12, the ratio is 12 divided by 40, which is 0.30, and the calculator displays it as 0.30 : 1 with the label overdrive. The output speed is 3,000 divided by 0.30, which is 10,000 RPM, more than three times the input, and the output torque is 10 times 0.30, which is only 3.00 N·m. An overdrive gear does not make a motor stronger; it makes a fast, weak output from a slower, stronger input, and it only makes sense when the load is light and the speed is the point, as in a top gear on a road vehicle. If you need torque, overdrive is the wrong direction, and the table shows why in one glance: the torque column collapses as the speed column grows.

Efficiency is the correction that turns the paper trade into a real one, and this page applies it to torque only, which is the honest place for it. Leave the efficiency at 100 percent and the default pair gives 33.33 N·m. Drop it to 95 percent, as the guide on this page discusses for a real mesh, and the output torque becomes 33.33 times 0.95, which is 31.67 N·m, while the output speed stays at 900 RPM. The calculator displays the torque multiplier as 3.17 rather than 3.33 in that case, which is 3.33 times 0.95, rounded to two decimals. Speed is set by the teeth counts alone, because meshing teeth cannot slip past each other without breaking the mesh; torque is where friction takes its cut, and it takes it as a percentage of the torque, not as a fixed amount. A motor chosen on the 100-percent torque figure, with no allowance for mesh losses, is a motor chosen slightly too small, and the efficiency field exists so that you do not have to make that mistake.

One more property belongs in this section because it surprises everyone who has not handled gears: direction. A single external mesh reverses the rotation, so the output of the default pair turns the opposite way to the input, and the calculator labels it reversed, one mesh. Direct drive through a single mesh still reverses, even though the speed and torque are unchanged: 40 driving 40 gives 3,000 RPM out and 10.00 N·m out, but in the opposite direction. Two meshes in series reverse twice and therefore restore the original direction, which is one reason compound trains are built the way they are. Direction never changes the ratio arithmetic, but it changes what the output shaft can be connected to, and it is worth checking before you commit to a layout rather than after.

Compound stages and bicycle gears: the two loaded examples

The page loads with two further examples already computed, one compound train and one bicycle setup, and both are worth reading carefully because they show what happens when the single-pair arithmetic is applied more than once. The compound example is a 3:1 first stage followed by a 2.5:1 second stage, from the loaded teeth counts of 12 driving 36 and then 10 driving 25. Stage ratios multiply, they do not add: 3 times 2.5 is 7.50, and the calculator displays a 7.50:1 total reduction with an output speed of 400 RPM from a 3,000 RPM input, which is 3,000 divided by 7.50. If you added the stages instead, you would get 5.50:1 and an output of about 545 RPM, which is wrong by more than a third, and the error grows with every stage you add. The practical lesson is that two modest reductions in series beat one extreme reduction on a single pair, because a single pair with a ratio of 7.50:1 would need a driven wheel seven and a half times the driver’s teeth, with all the size and tooth-loading problems that implies, while two stages of 3:1 and 2.5:1 keep both pairs at manageable sizes.

The bicycle example, also loaded on the page, is a 44-tooth chainring driving an 11-tooth rear cog at a cadence of 90 RPM, with a 27-inch wheel. The gear ratio is 44 divided by 11, which is 4.00:1, the rear wheel speed is 90 times 4, which is 360 RPM, and the gear inches figure is 108.0, which is the wheel diameter, 27 inches, times the ratio, 4.00. Gear inches are a way of expressing a bicycle gear as the diameter of an equivalent direct-drive wheel: every pedal turn rolls the bicycle as far as a 108-inch wheel would roll in one turn. The road speed displayed for this setup is 28.92 mph, with 46.54 km/h shown alongside, and the two are the same speed in different units, 28.92 times 1.60934 giving 46.54. Change the cadence or the cog and the whole chain of figures moves together, because the ratio, the wheel speed and the road speed are all proportional to each other once the wheel size is fixed. A climbing gear of 34 driving 28, which the page offers as a preset, has a ratio of about 1.21:1 and therefore gives a much lower road speed at the same cadence, trading speed for the torque a climb demands, exactly as the reduction rows in the table above do.

Both loaded examples also illustrate the limits of this page, stated honestly. The compound example assumes the stages are independent and that the efficiency figure applies overall, as the page’s own efficiency discussion explains; a real train has its own losses per mesh, its own tooth strengths and its own shaft layouts, and none of those are modelled here. The bicycle example assumes a single wheel size and a steady cadence, and it says nothing about wind, gradient, rolling resistance or the rider, all of which decide the speed a real cyclist actually reaches. What the page gives, in both cases, is the exact kinematic answer: the ratio, the speeds and the torque multiplication that the teeth counts alone determine. Everything beyond kinematics, strength, wear, lubrication and the load itself, belongs to the machine’s design, not to its gear ratio, and the page is careful not to pretend otherwise.

Teeth, diameters and the by-diameter mode

The calculator offers two ways of entering the same pair, by teeth and by diameter, and they are interchangeable only when the two wheels share the same tooth size, or module. By teeth, the ratio is driven teeth divided by driver teeth, exactly as in the table above. By diameter, the ratio is the driven diameter divided by the driver diameter, and the loaded example, a 2-inch driver and a 6-inch driven wheel, gives the same 3.00:1 family of answers as a teeth pair in the same proportion, because for wheels of the same module the diameter is proportional to the tooth count. The page’s diameter unit selector, inches or millimetres, changes the display, not the ratio, because both diameters are converted to the same unit before dividing. A pair entered as 50.8 mm driving 152.4 mm is the same pair as 2 inches driving 6 inches, and the ratio comes out the same either way.

Where the two modes diverge is in what they hide. Teeth counts are exact integers, so a ratio from teeth is exact up to the display rounding: 40 divided by 12 is 3.3333, displayed as 3.33, and the output speed of 900 RPM is exact because 3,000 divided by 3.3333 is 900. Diameters, by contrast, are measurements, and a measurement rounded to the nearest millimetre or the nearest sixteenth of an inch carries its rounding into the ratio. A driven wheel measured as 152 mm when it is truly 152.4 mm gives a ratio a fraction of a percent low, and at high speeds that fraction is a real number of RPM. The habit to take from this is simple: use teeth whenever you have them, because teeth are counted, not measured, and use diameters only when the teeth are inaccessible, as on a worn wheel or a belt pulley, and then measure both wheels with the same instrument and the same care. The page accepts both, but it cannot make a measured diameter as exact as a counted tooth.

A final check, useful whenever a ratio looks wrong, is to run the pair backwards. Swap driver and driven and the ratio inverts: 12 driving 40 gives 3.33:1, while 40 driving 12 gives 0.30:1, and the two output speeds at 3,000 RPM input, 900 and 10,000, are related by the same inversion, since 3,000 divided by 3.33 is 900 and 3,000 divided by 0.30 is 10,000. If your computed output speed is higher than your input speed when you expected a reduction, you have almost certainly entered the pair backwards, and the fix is to swap the two fields rather than to distrust the arithmetic. The calculator’s character labels, reduction, overdrive and direct drive, are there to catch exactly that mistake at a glance, before the numbers go any further into a design.