Formula and Method for Gear Ratio and Output Speed
When two gears mesh, their teeth pass across the point of contact at the same rate — a tooth leaving the driver gear's mesh is matched by a tooth leaving the driven gear's mesh at the same instant. That physical constraint means the number of teeth on each gear multiplied by its rotational speed is constant across the mesh: N₁ × ω₁ = N₂ × ω₂. From this, the gear ratio is defined as GR = N₂ / N₁ (driven teeth ÷ driver teeth), and the driven (output) gear's speed is the driver's speed divided by that ratio: ω_out = ω_in / GR. This calculator applies that relationship directly to your tooth counts and input speed.
How the calculation works
Enter the number of teeth on the driver (input) gear and the driven (output) gear, then the driver's rotational speed in RPM, rad/s, or Hz (rev/s). The tool first converts your speed entry to RPM, computes the gear ratio as GR = N_driven / N_driver, and finds the output speed as ω_out = ω_in × (N_driver / N_driven) = ω_in / GR. It also reports the output speed in rad/s (RPM × 2π/60) and the percent change between input and output speed, so you can see at a glance whether the driven gear runs faster or slower than the driver.
Common mistakes
- Swapping driver and driven: which gear you call the "driver" (the one supplying rotation) determines the direction of the ratio — reversing them inverts both the ratio and the speed/torque result.
- Ignoring idler gears: a simple idler meshed between the driver and driven gear reverses rotation direction but does not itself change the overall ratio — only the tooth counts of the first and last gears in the train set the ratio.
- Assuming zero losses: the ideal torque multiplier T_out ≈ T_in × GR ignores friction and bearing losses; real gearboxes typically deliver a few percent less torque than the ideal ratio predicts.
- Mixing speed units: RPM, rad/s, and Hz are not interchangeable without conversion — pick the unit that matches your input speed before reading the output.
Real-world applications
- Automotive transmissions and differentials trade engine RPM for wheel torque using stacked gear ratios.
- Bicycle drivetrains use chainring and cog tooth counts to set the relationship between pedaling cadence and wheel speed.
- Industrial gearboxes and gear reducers step down motor speed to match a driven load while multiplying available torque.
- Clock and watch gear trains chain several ratios together to convert a fast escapement oscillation into a slow, steady hour-hand rotation.