Hong Life CalculatorsUnderstanding the gear ratio calculator — concept and principle
When two spur gears mesh, the ratio of their tooth counts fixes how much the speed drops and the torque rises. This calculator gives the gear ratio, output speed, ideal output torque, and, from the module, the pitch diameters and centre distance you need to lay out the gears. It is used for gearbox sketches, robotics, hobby machines and checking replacement gears.
How it is calculated
- Gear ratio: i = z₂ / z₁ (driven teeth over driver teeth).
- Output speed: n₂ = n₁ × z₁ / z₂.
- Ideal output torque: T₂ = T₁ × z₂ / z₁ (no friction losses).
- Pitch diameter: d = m × z, where m is the module in mm.
- Centre distance: a = m × (z₁ + z₂) / 2.
The principle
Meshing teeth cannot slip, so both gears move the same pitch-circle length per second. Each tooth occupies the same arc on both gears, so one turn of the driver advances z₁ teeth, which turns the driven gear z₁/z₂ of a revolution. Power is torque times angular speed, and an ideal mesh transmits power unchanged, so if speed falls by a factor i the torque rises by i. The module is the pitch diameter per tooth; gears can only mesh if they share the same module and pressure angle (usually 20°). Since each pitch circle has radius m·z/2 and the circles touch, the centre distance is the sum of the two radii.
Worked example
- Inputs: driver 20 teeth, driven 60 teeth, 1,500 rpm, 50 N·m, module 2 mm.
- Ratio = 60 / 20 = 3.
- Output speed = 1,500 × 20 / 60 = 500 rpm.
- Output torque = 50 × 3 = 150 N·m.
- Pitch diameters = 2 × 20 = 40 mm and 2 × 60 = 120 mm.
- Centre distance = 2 × (20 + 60) / 2 = 80 mm.
Check: power in = 50 × 1,500 × 2π/60 ≈ 7.85 kW, power out = 150 × 500 × 2π/60, the same.
| Driver teeth | Gear ratio |
|---|---|
| 5 | 12 |
| 10 | 6 |
| 15 | 4 |
| 20 | 3 |
| 25 | 2.4 |
| 30 | 2 |
| 40 | 1.5 |
How to read the result
A ratio above 1 is a reduction (slower, more torque); below 1 is an overdrive. A single spur stage is commonly kept below roughly 5 to 7:1 for reasonable gear sizes; larger ratios use several stages, whose ratios multiply. Real torque is slightly lower than ideal because each spur mesh loses about 1–2% of power. For standard full-depth teeth the outside diameter is m(z + 2), so the gears above are 44 mm and 124 mm across.
Common mistakes
- Swapping driver and driven teeth, which inverts the ratio.
- Mixing module (mm) with diametral pitch (teeth per inch), which are reciprocal systems.
- Using outside diameter instead of pitch diameter for the centre distance.
- Choosing too few teeth: standard 20° gears below 17 teeth tend to undercut unless profile-shifted.
Frequently asked questions
Does an idler gear change the ratio?
No. An idler between two gears only reverses direction; the overall ratio is still z_output / z_input.
What are standard module values?
ISO 54 lists preferred modules such as 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 and 10 mm.
Why use a tooth count with no common factor?
A so-called hunting tooth ratio makes every tooth meet every other tooth over time, spreading wear more evenly.
Sources: ISO 54, Cylindrical gears for general and heavy engineering — Modules · Budynas & Nisbett, Shigley's Mechanical Engineering Design, McGraw-Hill