What is the gear ratio?
Enter the teeth on the driving and driven gears and the input speed. The calculator gives the gear ratio and the output speed, or any missing value.
- Output speed (rpm)
- 600
A 12-tooth gear driving a 36-tooth gear is a ratio of 3, so 1,800 rpm in gives 600 rpm out.
- Gear ratio
- 3
Output speed (rpm): 600. A 12-tooth gear driving a 36-tooth gear is a ratio of 3, so 1,800 rpm in gives 600 rpm out.
How to calculate
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.
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.
Formula: z₁ × n₁ = z₂ × n₂ (teeth × rpm is the same on both gears); ratio = z₂ ÷ z₁; n₂ = n₁ ÷ ratio.
- 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
Each example is checked against the calculator on every build.
- Driving gear teeth 12, Driven gear teeth 36, Input speed (rpm) 1,800 gives Gear ratio 3, Output speed (rpm) 600.Source: hand calculation in content.mdx: ratio = 36 ÷ 12 = 3 (3:1); output = 1,800 ÷ 3 = 600 rpm
- Driving gear teeth 50, Driven gear teeth 20, Input speed (rpm) 90 gives Gear ratio 0.4, Output speed (rpm) 225.Source: hand calculation in content.mdx: ratio = 20 ÷ 50 = 0.4; the wheel turns at 90 ÷ 0.4 = 225 rpm
- Driving gear teeth 15, Input speed (rpm) 3,000, Output speed (rpm) 750 gives Driven gear teeth 60, Gear ratio 4.Source: hand calculation in content.mdx: driven teeth = 15 × 3,000 ÷ 750 = 60; ratio = 60 ÷ 15 = 4
How it works
Two meshing gears move the same number of teeth past the point where they touch, every minute. So the teeth times the speed is the same on both gears: z₁ × n₁ = z₂ × n₂. From this:
- gear ratio = z₂ ÷ z₁ (driven teeth ÷ driving teeth)
- output speed n₂ = n₁ ÷ ratio
- input speed n₁ = n₂ × ratio
Here z₁ is the number of teeth on the driving gear, z₂ on the driven gear, and n₁ and n₂ are their speeds in revolutions per minute (rpm). The same holds for two sprockets joined by a chain, or two pulleys joined by a belt (with diameters in place of teeth).
Fill in any three of the four values z₁, z₂, n₁ and n₂, and the calculator finds the fourth from z₁ × n₁ = z₂ × n₂. It then shows the ratio z₂ ÷ z₁.
A ratio above 1 is a reduction: the output turns slower and, ideally, with more torque (output torque = input torque × ratio). A ratio below 1 is an overdrive.
Assumptions
- One pair of gears meshes directly. For a train of pairs, multiply the ratios of the pairs.
- No slip, and no friction losses.
- A solved tooth count can be a fraction, for example 10 teeth at 75 rpm driving a gear at 100 rpm gives 10 × 75 ÷ 100 = 7.5 teeth. Round it to a whole number of teeth and recheck the speed.
- Tooth counts are from 1 to 10,000 and speeds are above 0 and up to 10,000,000 rpm. If a solved value falls outside that range, there is no answer.
Worked examples by hand
A 12-tooth gear drives a 36-tooth gear at 1,800 rpm. Ratio = 36 ÷ 12 = 3 (3:1). Output speed = 1,800 ÷ 3 = 600 rpm.
A bicycle with a 50-tooth chainring and a 20-tooth rear sprocket, at 90 rpm. Ratio = 20 ÷ 50 = 0.4. The rear wheel turns at 90 ÷ 0.4 = 225 rpm.
The driven gear for 3,000 rpm to 750 rpm with a 15-tooth pinion. Driven teeth = 15 × 3,000 ÷ 750 = 60. Ratio = 60 ÷ 15 = 4.
Other questions people ask
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.