Gear Ratio Formula
Calculate gear ratio from teeth count or shaft speeds.
Understand torque multiplication and speed reduction with worked examples.
The Formula
Gear Ratio = Speed of Driving Gear / Speed of Driven Gear
The gear ratio tells you how many times the driving gear must rotate for the driven gear to rotate once. A ratio greater than 1 reduces speed but increases torque. A ratio less than 1 increases speed but reduces torque.
Variables
| Symbol | Meaning |
|---|---|
| GR | Gear ratio (dimensionless) |
| N₁ | Number of teeth on the driving gear (input) |
| N₂ | Number of teeth on the driven gear (output) |
| ω₁ | Rotational speed of driving gear (RPM) |
| ω₂ | Rotational speed of driven gear (RPM) |
Torque Relationship
Output Speed = Input Speed / Gear Ratio
Example 1
A driving gear has 20 teeth and meshes with a driven gear of 60 teeth. What is the gear ratio?
GR = N₂ / N₁ = 60 / 20
GR = 3:1 (the driven gear turns 3 times slower with 3 times more torque)
Example 2
A motor spins at 3,000 RPM with a gear ratio of 5:1. The input torque is 10 N·m. Find the output speed and torque.
Output speed = 3,000 / 5 = 600 RPM
Output torque = 10 × 5 = 50 N·m
Output: 600 RPM at 50 N·m (slower but much stronger)
When to Use It
Use the gear ratio formula in mechanical design:
- Designing gearboxes and transmissions for vehicles
- Selecting gear pairs for speed reduction or torque multiplication
- Calculating bicycle gear ratios for different terrains
- Sizing motors and gears for robotics and machinery
Key Notes
- Gear ratio: GR = N_driven / N_driving = ω_in / ω_out: N is the number of teeth (or sprocket teeth for chains); ω is angular velocity. A gear ratio > 1 means the output turns slower than the input (speed reduction, torque multiplication). Ratio < 1 means speed increase, torque reduction.
- Speed-torque trade-off: τ_out = τ_in × GR × η: Torque is multiplied by the gear ratio (minus efficiency losses η). A car in a low gear has a high gear ratio — the engine's torque is multiplied, enabling hill climbing and acceleration from rest at the cost of road speed.
- Compound gear trains: When multiple gear pairs are in series, total ratio = GR₁ × GR₂ × … × GRₙ. This allows very large overall ratios (clock mechanisms achieve millions to one) using compact stages of modest individual ratios.
- Velocity ratio from belt/chain drives: For pulleys connected by a belt: ω₁/ω₂ = D₂/D₁ (ratio of diameters, not teeth). Chains and gears use tooth counts; belts use diameters. The underlying principle — constant surface speed at the contact — is the same.
- Applications: Gear ratios are central to automotive transmissions, bicycle drivetrains, wind turbine gearboxes, industrial machinery, clockwork mechanisms, and robotic joint actuators where precise speed and torque conversion between motor and load are required.