Gear Ratio Calculator | RPM, Torque & Trains
Calculate gear ratio, output RPM, torque multiplier, and speed change for simple gear pairs and multi-stage gear trains. Visual SVG diagram included.
Quick presets
Stage 1
What Is the Gear Ratio Calculator | RPM, Torque & Trains?
A gear ratio describes the relationship between the number of teeth on two meshing gears, or equivalently, their pitch diameters. Because the gears mesh without slipping, the product of tooth count and rotational speed is constant at the meshing point: N&sub1;ω&sub1; = N&sub2;ω&sub2;. This meshing condition is the foundation of all gear calculations.
In a simple gear pair, a ratio greater than 1 means the driven gear is larger than the driver gear. The driven gear rotates more slowly but with proportionally greater torque, assuming ideal (lossless) transmission. Real-world efficiency for well-lubricated spur gears is typically 97–99% per stage; worm gears can be as low as 50%.
The gear train mode calculates compound ratios by multiplying all stage ratios together. The SVG diagram scales gear circles proportionally to tooth counts, giving a visual sense of relative sizes. All calculations run in the browser.
Formula
How to Use
- 1Choose a mode: Select "Simple gear pair" for a single mesh, or "Gear train" for up to 4 stages in series.
- 2Load a preset (optional): Click Bicycle 44/16, Car differential, Clock mechanism, Motorcycle gearbox, or Wind turbine to populate all fields instantly.
- 3Enter tooth counts: For each stage, enter the driver (input) gear tooth count and the driven (output) gear tooth count. Both must be positive whole numbers.
- 4Add more stages (train mode): Click "+ Add stage" to add up to 4 stages. The compound ratio is the product of all individual stage ratios.
- 5Enter input RPM: Type the rotational speed of the input (driver) shaft in revolutions per minute.
- 6Enter input torque: Type the torque applied to the input shaft in Newton-metres (N·m).
- 7Press Calculate: Results appear showing the gear ratio, output RPM, torque multiplier, speed change percentage, and rotation direction.
- 8Review the gear diagram: In simple mode, a proportional SVG diagram shows both gears scaled to their tooth counts. Expand step-by-step working to see each calculation substituted with your numbers.
Example Calculation
Example 1, Bicycle drivetrain
Chainring: 44 teeth at 90 RPM • Sprocket: 16 teeth
Gear ratio = 16 / 44 = 0.364:1
Output RPM = 90 / 0.364 = 247.5 RPM (wheel faster than pedals)
Speed increase = (247.5 - 90) / 90 = +175%
Torque multiplier = 0.364 (wheel torque is 36.4% of pedal torque)
Example 2, Automotive differential (3.73:1)
Engine at 3000 RPM • 200 N·m • Ring: 37 teeth, Pinion: 10 teeth
Gear ratio = 37 / 10 = 3.7:1
Output RPM = 3000 / 3.7 = 810.8 RPM (wheel speed)
Output torque = 200 × 3.7 = 740 N·m (ideal)
Speed reduction = (810.8 - 3000) / 3000 = -73%
Example 3, Two-stage gear train
Stage 1: 20 driver / 60 driven • Stage 2: 15 driver / 45 driven
Stage 1 ratio = 60 / 20 = 3.0
Stage 2 ratio = 45 / 15 = 3.0
Compound ratio = 3.0 × 3.0 = 9.0:1
At 1800 RPM input: Output = 1800 / 9 = 200 RPM
Torque multiplied by 9x (ideal)
Understanding Gear Ratio | RPM, Torque & Trains
Gear Ratio Fundamentals
Gears are simple machines that transfer rotational motion between shafts while transforming speed and torque. The gear ratio is determined entirely by the number of teeth (or pitch diameter) of the two meshing gears. No matter the material, tooth profile, or centre distance, the ratio is purely a function of tooth counts.
The law of conservation of energy applies to ideal gears: power in = power out. Since power = torque × angular velocity, halving the speed means doubling the torque. Real gears are 97–99% efficient per stage, so losses are small but non-zero.
Simple vs Compound Gear Trains
| Type | Description | Typical ratio range | Example |
|---|---|---|---|
| Simple pair | One driver, one driven gear | 1:1 to ~10:1 | Clock face gears |
| Compound train | 2+ stages on same shafts | Up to 100:1+ | Automotive gearbox |
| Planetary | Sun/planet/ring gears | 3:1 to 12:1 per stage | Automatic transmission |
| Worm gear | Crossed-axis high reduction | 5:1 to 100:1+ | Steering box, elevator |
| Rack & pinion | Rotary to linear motion | N/A (converts type) | CNC machine, steering |
Real-World Gear Ratio Examples
- Bicycle gearing: A 44-tooth chainring with a 16-tooth sprocket gives a 2.75:1 ratio, each pedal revolution moves the wheel 2.75 rotations. Shifting to a larger sprocket (lower gear) reduces the ratio for climbing.
- Automotive transmission: 1st gear is typically 3–4:1 for maximum torque at low speeds. 6th gear (overdrive) is often 0.6–0.8:1, reducing engine speed on highways to save fuel.
- Clock mechanism: A 12-hour clock needs the minute hand shaft to turn 12 times per hour-hand revolution. This is achieved by a compound gear train with an overall 12:1 ratio.
- Wind turbines: Generators require 1000–1500 RPM while rotor blades turn at 10–20 RPM, requiring overall ratios of 50:1 to 100:1, typically using a multi-stage helical gearbox.
Frequently Asked Questions
What does a gear ratio of 3:1 mean exactly?
A 3:1 ratio means the driver gear rotates three times for every one rotation of the driven gear. Equivalently:
- • Output speed = input speed / 3
- • Output torque = input torque × 3 (ideal, no losses)
- • The driven gear has 3× as many teeth as the driver
This is a speed reduction / torque multiplication arrangement, typical of automotive low gears and industrial gearboxes.
Why do external gears rotate in opposite directions?
When two external gears mesh, their teeth push against each other at the contact point, causing them to rotate in opposite directions. Each stage in a gear train reverses direction:
- • 1 stage: output is opposite to input
- • 2 stages: output is same as input
- • n stages: same if n is even, opposite if n is odd
Internal (ring) gears rotate in the same direction as the driving gear, used in planetary gearsets.
How does mechanical efficiency affect the output torque?
This calculator shows ideal output torque (100% efficiency). Real gear systems have losses:
- • Spur/helical gears: 97–99% per stage
- • Bevel gears: 97–99%
- • Worm gears: 50–90% (high reduction ratios are very lossy)
- • Planetary gears: 97–99%
Multiply the ideal torque by the efficiency factor to get realistic output torque: T_out = T_in × ratio × efficiency
Can I use this for belt-and-pulley systems?
Yes. Belt-and-pulley systems follow the same mathematical relationship as gear trains:
Ratio = Driven pulley diameter / Driver pulley diameter
For V-belts and timing belts, use pitch diameter (not outer diameter). The tooth count on toothed (synchronous) belts is equivalent to gear teeth for exact ratio calculation.
What is a compound gear train?
A compound gear train has multiple gear meshes in series on the same shaft. The overall ratio is the product of all individual stage ratios. This allows very high or very low total ratios in a compact space.
Example: three stages of 3:1 each give a compound ratio of 3×3×3 = 27:1. A single gear pair achieving 27:1 would require an impractically large driven gear.
What is a gear ratio in terms of angular velocity?
The meshing condition states that at the contact point, both gears have the same linear velocity:
ω&sub1; × r&sub1; = ω&sub2; × r&sub2; (where r is pitch radius)
Since pitch radius is proportional to tooth count, this gives: N&sub1;ω&sub1; = N&sub2;ω&sub2;, which rearranges to the gear ratio formula. Angular velocity (ω) and RPM are proportional, so the formula works identically in either unit.
How do I size gears for a specific output speed?
To achieve a target output speed from a known input speed:
- 1. Calculate required ratio: ratio = Input RPM / Target RPM
- 2. Choose a driver tooth count (e.g. 20 teeth)
- 3. Driven teeth = driver teeth × ratio (e.g. 20 × 3 = 60 teeth)
- 4. If the ratio is not achievable with a single stage, split across multiple stages
Always choose standard tooth counts from gear catalogues for interchangeability.
What is the difference between gear ratio and overall drive ratio?
A vehicle's overall drive ratio is the product of the transmission gear ratio and the axle/differential ratio. For example:
- • Transmission 1st gear: 3.5:1 • Differential: 3.73:1
- • Overall 1st gear ratio = 3.5 × 3.73 = 13.1:1
- • Engine at 2000 RPM → wheel speed = 2000 / 13.1 = 152.7 RPM
Use the gear train mode to chain transmission and differential ratios together.