Maths⏱ 5 min read
How Gear Ratios Work: Calculating Mechanical Advantage
Gear ratios determine how force is amplified or speed is increased in any mechanical system. Here's the formula for simple gear trains, bicycle gearing, and cars — with practical examples.
Gear ratios appear everywhere from bicycles and cars to industrial machinery and watches. The underlying maths is the same in every case: trading force for speed or speed for force.
The Basic Gear Ratio
Gear Ratio = Driven gear teeth / Driving gear teeth
(or equivalently: Input speed / Output speed)
A 60-tooth gear driven by a 20-tooth gear:
Ratio = 60 / 20 = 3:1
This means:
- The input shaft rotates 3 times for every 1 output rotation
- Output torque = Input torque x 3 (force is amplified)
- Output speed = Input speed / 3 (speed is reduced)
Fundamental trade-off: you can amplify force OR speed, not both.
Energy (power) is conserved minus friction losses.
Bicycle Gearing
Bicycle gear ratio = Chainring teeth / Sprocket teeth
Common gravel/road bike example:
Chainring: 50 teeth
Sprocket (rear): 12 teeth
Ratio: 50 / 12 = 4.17
This means 4.17 rear wheel rotations per pedal revolution.
Development (distance per pedal stroke):
Development = (Chainring / Sprocket) x Wheel circumference
Wheel circumference (700c road): 2.1m
Development = 4.17 x 2.1 = 8.76m per pedal stroke
At 90 RPM cadence:
Speed = 8.76m x 90 / 60 = 13.14 m/s = 47.3 km/h
Easiest climbing gear (34 tooth chainring, 32 tooth sprocket):
Ratio = 34/32 = 1.06 (almost 1:1)
Development = 1.06 x 2.1 = 2.23m per pedal stroke
Speed at 90 RPM = 2.23 x 90/60 = 3.34 m/s = 12 km/h
Gear Ratio in Cars (Transmission)
Overall drive ratio = Gearbox ratio x Final drive (differential) ratio
Example: 5-speed manual gearbox
1st gear: 3.36 x final drive 3.75 = 12.6:1 total (high force, low speed)
2nd gear: 1.99 x 3.75 = 7.46:1
3rd gear: 1.36 x 3.75 = 5.10:1
4th gear: 1.00 x 3.75 = 3.75:1 (direct drive)
5th gear: 0.82 x 3.75 = 3.08:1 (overdrive — cruising efficiency)
Engine at 2,000 RPM in 5th gear, wheel diameter 0.63m:
Wheel RPM = 2,000 / 3.08 = 649 RPM
Wheel circumference = pi x 0.63 = 1.979m
Speed = 649 x 1.979 / 60 = 21.4 m/s = 77 km/h
Compound Gear Trains
When multiple gear pairs are in series, multiply ratios together.
Gear A (20 teeth) drives Gear B (60 teeth): ratio 3:1
Gear B shaft also has Gear C (15 teeth) driving Gear D (45 teeth): ratio 3:1
Overall ratio = 3 x 3 = 9:1
This is how watches achieve ratios of thousands to one —
multiple small gear stages stacked together.
Planetary (epicyclic) gears used in automatic transmissions
achieve multiple ratios from the same set of gears by locking
different elements — more compact than stacked spur gears.