Gear Ratio Calculator

Compute gear ratio from teeth or diameters, see how it changes speed and torque, and evaluate multi‑stage drivetrains and vehicle speed from engine RPM and tire size.

1. Single Gear Pair

Use diameters only if teeth are unknown. Teeth and diameter methods should give similar ratios.

Used for torque output estimate.

Gear ratio basics

Gear ratio tells you how the speed and torque change between an input gear and an output gear. It is defined as:

From teeth:

Gear ratio \( R = \dfrac{T_\text{driven}}{T_\text{driving}} \)

From pitch diameters:

Gear ratio \( R = \dfrac{D_\text{driven}}{D_\text{driving}} \)

If \(R = 3\), we usually say “3:1 ratio” – the driven gear turns once for every three turns of the driving gear.

Speed and torque relationships

Ignoring losses, power is conserved: \(P = \tau \cdot \omega\). This leads to:

Output speed: \( n_\text{out} = \dfrac{n_\text{in}}{R} \)

Output torque (ideal): \( \tau_\text{out} = \tau_\text{in} \cdot R \)

With efficiency \( \eta \) (0–1): \( \tau_\text{out} \approx \tau_\text{in} \cdot R \cdot \eta \)

  • High ratio (e.g. 4:1): lower speed, higher torque – good for climbing or heavy loads.
  • Low ratio (e.g. 1:1 or 0.8:1): higher speed, lower torque – good for cruising.

Overall drive ratio in multi‑stage systems

In a gearbox, bicycle, or robot, you often have several stages in series. The overall ratio is the product of all stage ratios:

\( R_\text{overall} = R_1 \times R_2 \times R_3 \times \dots \)

Example: 1st gear 3.5:1 and final drive 4.1:1 → \( R_\text{overall} = 3.5 \times 4.1 \approx 14.35:1 \).

The calculator’s multi‑stage mode does this automatically and gives you the final speed and torque.

Vehicle speed from gear ratio and tire size

For vehicles, speed depends on wheel RPM and tire circumference:

Wheel RPM: \( n_\text{wheel} = \dfrac{n_\text{engine}}{R_\text{gear} \cdot R_\text{final}} \)

Tire circumference: \( C = \pi \cdot d \)

Speed (m/s): \( v = \dfrac{n_\text{wheel} \cdot C}{60} \)

Convert to km/h: \( v_\text{km/h} = v \cdot 3.6 \)

Convert to mph: \( v_\text{mph} = v \cdot 2.23694 \)

The vehicle tab handles unit conversions for tire diameter and reports both mph and km/h.

Worked example

Single pair

Driving gear: 20 teeth, driven gear: 60 teeth, input 1500 RPM, 100 N·m, 95% efficiency.

  • Ratio \( R = 60 / 20 = 3:1 \).
  • Output speed \( n_\text{out} = 1500 / 3 = 500 \) RPM.
  • Ideal torque \( = 100 \times 3 = 300 \) N·m.
  • With 95% efficiency: \( 300 \times 0.95 = 285 \) N·m.

Vehicle example

Engine 3000 RPM, 4th gear 1.00:1, final drive 3.90:1, tire diameter 26 in.

  • Overall ratio \( R_\text{overall} = 1.00 \times 3.90 = 3.90:1 \).
  • Wheel RPM \( = 3000 / 3.90 \approx 769.2 \) RPM.
  • Tire circumference \( C \approx \pi \times 26\text{ in} \approx 81.68\text{ in} \approx 2.074\text{ m} \).
  • Speed \( v = 769.2 \times 2.074 / 60 \approx 26.6\text{ m/s} \approx 95.8\text{ km/h} \approx 59.5\text{ mph} \).

Tips for choosing gear ratios

  • Robotics / machinery: start from required output torque and speed, then back‑calculate the ratio from motor specs.
  • Automotive: use the vehicle tab to see cruising RPM at highway speeds and acceleration in lower gears.
  • Bicycles: treat chainrings and cogs as gears; higher ratios give more speed but require more force.

FAQ