# What is the gear ratio?

Computes the gear ratio of a driving and a driven gear from their teeth, and the output speed in rpm from the input speed, or a missing teeth count or speed from the other three.

- Page: https://www.acalculator.org/physics/gear-ratio-calculator
- JSON spec: https://www.acalculator.org/physics/gear-ratio-calculator.json
- Version: fc6f49112ed8

## Default answer

Example with the default inputs (Driving gear teeth 12, Driven gear teeth 36, Input speed (rpm) 1,800): A 12-tooth gear driving a 36-tooth gear is a ratio of 3, so 1,800 rpm in gives 600 rpm out.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| z1 | Driving gear teeth | The number of teeth on the driving (input) gear. |
| z2 | Driven gear teeth | The number of teeth on the driven (output) gear. |
| n1 | Input speed (rpm) | The speed of the driving gear, in revolutions per minute. |
| n2 | Output speed (rpm) | The speed of the driven gear, in revolutions per minute. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| driver | Driving gear teeth | The number of teeth on the driving (input) gear. |
| driven | Driven gear teeth | The number of teeth on the driven (output) gear. |
| inputRpm | Input speed (rpm) | The speed of the driving gear, in revolutions per minute. |
| outputRpm | Output speed (rpm) | The speed of the driven gear, in revolutions per minute. |
| ratio | Gear ratio | Driven teeth divided by driving teeth: how many turns of the input make one turn of the output. |

## Method

z₁ × n₁ = z₂ × n₂ (teeth × rpm is the same on both gears); ratio = z₂ ÷ z₁; n₂ = n₁ ÷ ratio.

## Assumptions

- Two gears (or two sprockets on a chain) mesh directly. For a train of several pairs, multiply the ratios of the pairs.
- Tooth counts are from 1 to 10,000 and speeds up to 10,000,000 rpm. A solved value outside that range has no answer.
- The teeth do not slip, so the teeth that pass the contact point per minute are the same on both gears.
- The ideal output torque is the input torque times the ratio. Real gears lose a few percent to friction; this calculator does not include losses.

## Worked examples

1. z1 = 12, z2 = 36, n1 = 1,800 gives ratio = 3, n2 = 600. Source: hand calculation in content.mdx: ratio = 36 ÷ 12 = 3 (3:1); output = 1,800 ÷ 3 = 600 rpm.
2. z1 = 50, z2 = 20, n1 = 90 gives ratio = 0.4, n2 = 225. Source: hand calculation in content.mdx: ratio = 20 ÷ 50 = 0.4; the wheel turns at 90 ÷ 0.4 = 225 rpm.
3. z1 = 15, n1 = 3,000, n2 = 750 gives z2 = 60, ratio = 4. Source: hand calculation in content.mdx: driven teeth = 15 × 3,000 ÷ 750 = 60; ratio = 60 ÷ 15 = 4.

## FAQ

### How do I calculate a gear ratio?

Divide the number of teeth on the driven (output) gear by the number of teeth on the driving (input) gear. A 12-tooth gear driving a 36-tooth gear has a ratio of 36 ÷ 12 = 3, written 3:1.

### How does the gear ratio change the speed?

The output speed is the input speed divided by the ratio. With a 3:1 ratio, 1,800 rpm in gives 1,800 ÷ 3 = 600 rpm out. A ratio below 1 speeds the output up.

### How does the gear ratio change the torque?

In an ideal gear pair, the output torque is the input torque times the ratio, so a 3:1 reduction triples the torque while it cuts the speed to a third. Real gears lose a few percent of the power to friction.

### What about a gear train with more than two gears?

Multiply the ratios of each meshing pair. For example, a 3:1 pair followed by a 4:1 pair gives 12:1. An idler gear between two gears changes the direction of rotation but not the overall ratio.

### Does this work for bicycle gears and chains?

Yes. The chainring drives the rear sprocket, so the ratio is rear teeth ÷ front teeth. A 50-tooth chainring and a 20-tooth sprocket give 0.4, so the wheel turns 1 ÷ 0.4 = 2.5 times per turn of the pedals. Cyclists often quote the inverse, front ÷ rear = 2.5.

## Sources

- ISO 1122-1:1998, Vocabulary of gear terms, Part 1: Definitions related to geometry. Gear ratio u = z₂ ÷ z₁ and transmission ratio i = ω₁ ÷ ω₂. https://www.iso.org/standard/5725.html
- Budynas, R. G., and J. K. Nisbett. Shigley's Mechanical Engineering Design, 11th edition, McGraw-Hill (2020), chapter 13, Gears: general, section 13-12 (gear trains).