A gear pair is a trade: speed for torque, at an exchange rate set by the teeth. Count them, and everything else — output rpm, torque multiplication, whether the machine crawls or cruises — follows from one division. This gear ratio calculator takes the driving and driven tooth counts (or two pulley/gear diameters), your input speed and torque, and returns the ratio, the output speed and the output torque, losses aside.
The Ratio and Its Two Consequences
Gear ratio = driven teeth ÷ driving teeth. Meshing gears pass the same number of teeth per second at the contact, so speeds trade inversely to teeth: output rpm = input rpm ÷ ratio, and torque — power being (nearly) conserved — trades the other way: output torque ≈ input torque × ratio. The page’s default is the classic reduction: a 12-tooth pinion driving a 36-tooth gear gives ratio 3.00 : 1, turning 3,000 rpm in into 1,000 rpm out and 10 Nm in into about 30 Nm out. Reverse the pair (ratio 0.333) and the same hardware becomes an overdrive: triple the speed, a third of the torque. There is no free side of the trade — that is the entire physics, and the entire point.
Teeth, Diameters and Sprockets
Teeth are the honest input because they are countable; but pitch diameters divide identically (ratio = D₂/D₁) since teeth are proportional to diameter on meshing gears of shared module. The same arithmetic runs belt pulleys and chain sprockets (count sprocket teeth; measure pulley diameters — belts add a percent or two of slip and creep that gears, being toothed, refuse). Bicycle chainrings and cogs are the friendliest example on earth: with the 44-tooth ring driving an 11-tooth cog, the ratio (driven ÷ driving) is 11/44 = 0.25 — an overdrive spinning the wheel four times per pedal revolution. Mechanics quoting “a 4:1 bike gear” mean the same thing from the wheel’s side: read every quoted ratio for its convention before comparing.
Compound Trains and Idlers
Real gearboxes chain pairs, and total ratio is the product of the pairs: two 3:1 stages give 9:1, which is how a small motor lifts a car on a jack or a winch winds at a crawl. An idler gear between two gears changes nothing numerically — its teeth cancel from the product — but flips the output direction back to the input’s, which is its whole job (and why reverse gear in a manual box is an idler sliding in). Direction bookkeeping: every external mesh reverses rotation; count the meshes to know which way the output turns, a detail the calculator flags because assemblies have been built wrong on exactly that point.
Efficiency: The Tax on Every Mesh
The torque multiplication printed here is ideal. Each real spur or helical mesh returns about 95–98 percent of its input power (friction, churning oil, bearing losses); worm gears, trading sliding contact for compact huge ratios, may return only 50–80 percent and are often chosen because they cannot be back-driven. Chain drives sit near 95–98 percent, belts a touch lower with slip. Design practice applies the losses per stage — three 97-percent stages deliver 91 percent overall — and sizes the motor for the output torque plus the tax, a line item that separates machines that stall from machines that ship.
Where Ratios Rule
Vehicle drivetrains stack three ratios — gearbox, transfer case if fitted, final drive — and multiply them: a 3.6 first gear through a 3.9 final drive is 14:1 at the wheels, which is why first crawls and top (0.7 overdrive × 3.9 ≈ 2.7) cruises. Machine tools gear for torque at the cut; clocks gear almost purely for ratio accuracy (the teeth are the timekeeping); wind turbines gear up 1:50 or more to spin generators at useful speeds; robotics lives on harmonic and planetary reductions of 50–100:1 in a fist-sized joint. In every field the design conversation starts with the same division this calculator performs.
Common Mistakes
Inverting the ratio convention mid-calculation (decide driven ÷ driving once, as here, and never swap). Expecting torque and speed both to rise. Forgetting direction reversal per mesh. Mixing tooth counts with diameters from different module families — gears that do not share a module do not mesh, whatever their ratio. And neglecting efficiency for multi-stage trains, then blaming the motor for arithmetic it never agreed to.
Related Tools
The machines bench continues in the Physics Calculators hub — the Voltage Divider Calculator is gearing’s electrical twin, and the Speed of Sound Calculator covers wave speeds in the media gears sing through.
Frequently Asked Questions
How do you calculate gear ratio?
Divide the driven gear’s teeth by the driving gear’s teeth. A 36-tooth gear driven by a 12-tooth pinion gives 3:1 — output speed is the input divided by 3 and output torque is (ideally) tripled.
What does a 3:1 gear ratio mean?
The input shaft turns three times for each single turn of the output. Speed divides by three, torque multiplies by three (minus friction losses) — a reduction gearing, used wherever force matters more than speed.
Can gear ratio be calculated from diameters?
Yes — pitch diameters divide exactly like tooth counts, because teeth are proportional to diameter on meshing gears. Ratio = driven diameter ÷ driving diameter. The same holds for pulley diameters and sprocket teeth.
What is an overdrive gear?
Any pair with ratio below 1: the output spins faster than the input, paying in torque. Top gears in cars are overdrives (around 0.7:1) so the engine can loaf at highway speed.
Does an idler gear change the ratio?
No — its teeth cancel out of the product of stages. Its purpose is direction: it makes the output turn the same way as the input. Reverse gears in manual transmissions are idlers doing exactly that.
How much torque is lost in a gearbox?
Typically 2–5 percent per spur or helical mesh, compounding per stage; worm drives can lose 20–50 percent. Multiply the ideal output torque by about 0.95 per mesh for a working estimate.