Formula and Method for the Gear Ratio RPM Calculator
When two gears mesh, the smaller "driver" gear turns the larger "driven" gear (or vice versa) through direct tooth contact. Because meshed teeth move at the same linear speed at the point of contact, the gear with fewer teeth must rotate faster than the gear with more teeth. The gear ratio (GR) captures this relationship as the number of teeth on the driven gear divided by the number of teeth on the driver gear: GR = N₂ ÷ N₁. This calculator uses that ratio to convert an input speed and input torque into the resulting output speed and output torque.
How the calculation works
Enter the driver gear's tooth count (N₁), the driven gear's tooth count (N₂), the input speed in RPM, and — optionally — the input torque in newton-meters. The calculator first finds the gear ratio, GR = N₂ ÷ N₁. Output speed follows from the fact that speed and tooth count are inversely proportional: RPM_out = RPM_in × (N₁ ÷ N₂), which is the same as dividing the input speed by GR. Output torque assumes an ideal, frictionless mesh where mechanical power (torque × angular speed) is conserved across the gear pair, so torque scales up by the same factor speed scales down: T_out = T_in × GR. Real gearboxes run at roughly 90-98% efficiency, so measured output torque will be slightly below this ideal value once bearing and mesh friction are accounted for.
Common mistakes
- Swapping driver and driven: the gear ratio depends on which gear receives the power (driver, N₁) and which gear is turned by it (driven, N₂) — reversing the two inverts the ratio.
- Mixing teeth counts with diameters: gear ratio can also be computed from pitch diameters (GR = D₂ ÷ D₁), but do not mix a teeth count for one gear with a diameter for the other.
- Forgetting direction reversal: a single pair of external meshed gears always spins the output in the opposite direction from the input; an idler gear placed between them reverses direction again without changing the numeric ratio.
- Ignoring efficiency losses: the torque formula here is the ideal, lossless case — real output torque is somewhat lower once friction and windage losses are subtracted.
Real-world applications
- Automotive transmissions and differentials use gear ratios to trade engine speed for wheel torque at low gears and wheel speed for efficiency at high gears.
- Bicycle drivetrains use the ratio of chainring teeth to rear cog teeth to set how hard versus how fast the rider pedals.
- Industrial gear reducers step down a motor's high RPM to the slower, higher-torque speed a conveyor, mixer, or pump needs.
- Wind turbine gearboxes step up the rotor's slow rotation to the fast speed a generator requires to produce electricity efficiently.