TAHA Engineering Toolsby TAHA Knowledge Lab

Mechanical Engineering

Gear Ratio Calculator

Gear ratio from teeth or speeds, with output speed, torque and direction for a gear pair.

Inputs

Calculate from
Small driver gear with z1 teeth meshing with a larger driven gear with z2 teeth; they rotate in opposite directions.Driver z₁n₁, T₁ inDriven z₂n₂, T₂ out
An external pair: the driver gear (z₁ teeth) turns the driven gear (z₂ teeth) in the opposite direction. (A pinion inside a ring gear turns it the same way.) Ratio i = z₂ ÷ z₁; output speed n₂ = n₁ ÷ i. Diagram not to scale.

External: two ordinary spur or helical gears side by side. Internal: a small gear running inside a ring gear with inward-facing teeth.

100% means ideal (no friction losses). Enter a lower value if you know the efficiency of the gear pair.

Result

Gear ratio
3: 1
Output speed
500
Output torque
30
Output rotation
Opposite direction to the driver

The gear ratio is 3 : 1. This is a speed reduction: the output turns slower than the input and delivers more torque. Output: 500 rpm and 30 N·m, turning in the opposite direction to the driver.

Educational estimate — check the assumptions below.

About this calculator

The gear ratio compares the number of teeth on the driven gear with the driver. It tells you how a gear pair changes speed and torque: a reduction (ratio above 1) slows the output and multiplies torque, while an overdrive (ratio below 1) does the opposite. Power stays the same apart from friction losses, which you can include as a mesh efficiency.

Use From tooth counts when you know the teeth on both gears. Use From input and output speeds when you have measured the two shaft speeds, for example with a tachometer. The ratio is a size (magnitude) only: the direction of rotation is shown separately, because it depends on the type of gear pair.

Formula

From tooth counts

i = z2 ÷ z1

n2 = n1 × z1 ÷ z2 = n1 ÷ i    T2 = T1 × i × η

z1, z2 = teeth on driver and driven gear, n = speed, T = torque, η = mesh efficiency (as a fraction). External gears turn in opposite directions; a pinion and an internal (ring) gear turn in the same direction.

From input and output speeds

i = n1 ÷ n2    T2 = T1 × i × η

n1 = input speed, n2 = output speed (both in the same unit), T = torque, η = mesh efficiency (as a fraction).

Worked example

A 20-tooth pinion drives a 60-tooth gear. The pinion turns at 1500 rpm with 10 N·m of torque.

  • Ratio: i = 60 ÷ 20 = 3 : 1
  • Output speed: 1500 ÷ 3 = 500 rpm
  • Output torque (ideal, η = 100%): 10 × 3 = 30 N·m
  • With η = 97%: 10 × 3 × 0.97 = 29.1 N·m
  • Direction: external gears, so the output turns the opposite way to the input.

From speeds: a motor runs at 1450 rpm and the output shaft is measured at 290 rpm. Ratio i = 1450 ÷ 290 = 5 : 1.

Assumptions and limitations

  • One simple pair of meshing gears on fixed shafts. For a compound gear train, multiply the ratios of each pair. Idler gears change direction but not the overall ratio.
  • These formulas do not apply to planetary (epicyclic) gear sets, differentials or other gearboxes where shafts or carriers move; their ratios depend on which member is held.
  • The calculator does not check that the gears can actually mesh (same module or diametral pitch, correct centre distance).
  • Gear tooth strength, bearing loads and lubrication are not checked. Size real gearboxes using the manufacturer’s ratings.

References

  • Standard gear kinematics: speed ratio equals the inverse tooth ratio for meshing gears.