Gear Ratio & Speed Calculator

Calculate gear ratio, output speed, torque, and power transmission for gear systems.

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Gear Ratio Trades Speed for Torque

An ideal gear pair changes rotational speed and torque while transmitting approximately the same mechanical power. For external gears, the speed ratio is set by tooth counts or pitch diameters. If gear 1 drives gear 2, ω2/ω1=N1/N2. A larger driven gear rotates more slowly but can deliver proportionally greater torque in an ideal lossless system.

Meshing external gears rotate in opposite directions, while additional idler gears can change direction without changing the overall magnitude of the ratio. Real gearboxes have friction, tooth deformation, bearing losses, and sometimes multiple stages, so output power is lower by the efficiency factor.

i=Ndriven/Ndriver=ωdriver/ωdriven,   τout≈ηiτin
SymbolMeaningWhy it appears / units
NNumber of teethDimensionless; ratio of tooth counts sets speed ratio for meshing gears.
iGear ratioDimensionless; convention must be stated clearly.
ωAngular speedrad/s or rpm; output speed decreases for reduction ratios greater than 1.
ηEfficiency0 to 1; accounts for real losses.

A 4:1 reduction means the output turns once for four input revolutions and ideally produces about four times the torque. Power is not multiplied; increased torque is paid for by decreased speed, with additional losses in real systems.

Gear speed and torque should trade in opposite directions in an ideal pair. If the driven gear has more teeth than the driver, output speed falls and output torque rises, apart from losses. Pitch-diameter ratios should agree with tooth-count ratios for gears of the same module.

Worked Examples

Example 1: Speed reducer: 20→60 teeth, 1500RPM, T=10N·m
GR=3, outRPM=500
Result: T_out=30N·m — speed halved, torque tripled
Typical motor-to-machine gearbox
Example 2: Bicycle: chainring 52T, sprocket 13T, 90RPM pedaling
GR=13/52=0.25, wheel RPM=360
Result: 4× speed increase
High gear for fast cycling
Example 3: Speed reduction
20-tooth driver, 60-tooth driven, input=1800rpm
Result: i=3, output=600rpm
The larger driven gear turns at one-third the input speed.
Example 4: Torque with efficiency
i=3, input torque=10N·m, η=0.90
Result: output torque≈27N·m
Ideal torque would be 30N·m; 90% efficiency reduces the transmitted output.

Common Mistakes

⚠️
Reversing driver and driven tooth counts

State the ratio convention before calculating. Swapping the gears changes reduction into speed increase.

⚠️
Assuming gears create power

Ideal gears trade speed for torque while conserving power. Real gears lose some power to friction and other effects.

⚠️
Ignoring rotation direction

A single pair of external gears reverses rotation direction even though ratio magnitude is usually reported as positive.

Frequently Asked Questions

Torque-speed tradeoff?
Power = Tω = constant (ideal, no friction). Step-down gear: reduces speed, increases torque by same ratio. Step-up: increases speed, reduces torque. Efficiency: helical gears 97-99%, worm gears 40-85%.
Gear ratio selection?
Match motor peak efficiency speed to load requirements. Electric motors: typically 1000-3000 RPM. Final output: application-specific (machine tool: 100-1000 RPM, conveyor: 10-100 RPM, antenna: <1 RPM).
Does an idler gear change the gear ratio?
An ideal idler placed between driver and driven gears changes rotation direction or spacing but not the magnitude of the overall ratio, which still depends on the first and last gear tooth counts.
Can gear ratio be calculated from diameters?
Yes, for gears with compatible tooth pitch, the ratio of pitch diameters equals the ratio of tooth counts. Outside diameters should not be substituted blindly for pitch diameters.
Why does torque increase when speed decreases?
Mechanical power is P=τω. If power is approximately conserved, reducing angular speed allows torque to rise proportionally, apart from efficiency losses.
How do multiple gear stages combine?
Multiply the stage ratios to obtain the overall ratio. Multiply stage efficiencies as well when estimating real output torque and power.
Which gear ratio convention should I use?
For a simple external gear pair, a common speed-reduction convention is i=Ndriven/Ndriver, giving ωout=ωin/i. Other references define the reciprocal, so state the convention. External meshing also reverses rotation direction even when a magnitude-only calculator reports positive speeds.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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