Gear Ratio Calculator

What the gear ratio calculator does

When two gears mesh, the one with more teeth turns more slowly but with more turning force. This calculator works out the gear ratio from the tooth counts of the input and output gears, then shows how the rotational speed and the torque change from one to the other.

Enter the teeth on each gear plus the input speed and torque, and it gives the ratio along with the output speed and output torque. Below is the trade gears make, the equations behind it, and a worked example.

How to use it

1. Enter the number of teeth on the input gear and on the output gear.
2. Enter the input rotational speed in RPM and the input torque in newton-metres.
3. Press Calculate for the gear ratio, output speed, and output torque, or Reset to clear it.

Gears trade speed for turning force

Gears are a way of trading rotational speed for torque, the twisting force that turns a shaft. Mesh a small gear driving a large one and the large gear turns slower, but it turns harder. Mesh a large gear driving a small one and the small gear spins faster, but with less force. You choose which way to make the trade by choosing the sizes.

This is why a car has low gears and high gears. A low gear, with a big reduction, gives the wheels lots of torque to pull away from a stop or climb a hill, at the cost of speed. A high gear gives speed on the open road, at the cost of pulling power. The same idea lets a small motor in a power drill or a robot produce a strong, slow turn from a fast, weak one.

The equations it uses

The gear ratio is the number of teeth on the output gear divided by the number on the input gear:

ratio = output teeth ÷ input teeth

The output speed is the input speed divided by that ratio, and the output torque is the input torque multiplied by it:

output speed = input speed ÷ ratio and output torque = input torque × ratio

So a ratio above one, from a larger output gear, slows the rotation and multiplies the torque. A ratio below one, from a smaller output gear, speeds the rotation and reduces the torque. Speed and torque always move in opposite directions.

Why power stays the same

Notice that as the ratio raises the torque it lowers the speed by the same factor, and the other way round. That is not a coincidence. Rotational power is torque times rotational speed, and gears cannot create power, they can only pass it along. So whatever one quantity gains, the other must lose in proportion, keeping their product, the power, unchanged.

This is the rotational version of the bargain every machine makes: you can have more torque or more speed, but not both, because the power is fixed by what drives the input. A real gear train loses a little to friction, so the output power is slightly less than the input, but the trade itself is set by this rule. It is why you cannot gear your way to free power, only to the balance of speed and force that suits the job.

Units and precision

The gear ratio itself is a pure number, since it compares one tooth count to another. The calculator takes the input speed in RPM and returns the output speed in RPM, and takes torque in newton-metres and returns it the same way. Results are rounded to a few places for display while the calculation runs at full precision.

A worked example: a 3 to 1 reduction

Take an input gear of 12 teeth driving an output gear of 36 teeth, with the input turning at 1,500 RPM and supplying 10 N·m of torque.

The gear ratio is 36 ÷ 12 = 3. The output speed is 1,500 ÷ 3 = 500 RPM, a third of the input, and the output torque is 10 × 3 = 30 N·m, three times the input. Speed has been traded for torque in a three-to-one reduction, while the power, speed times torque, stays the same on both sides.

Questions people ask

How do you calculate gear ratio?

Divide the number of teeth on the output gear by the number on the input gear. A ratio of 3 means the output turns once for every three turns of the input.

Does a gear ratio change speed or torque?

Both, in opposite directions. A reduction ratio lowers the output speed and raises the torque by the same factor, while an overdrive ratio does the reverse.

Can gears increase power?

No. Gears trade speed for torque while keeping their product, the power, the same, minus a small loss to friction. They change the balance of speed and force, not the total power.

What is a gear reduction?

It is a gear pairing where the output gear is larger than the input, giving a ratio above one. It slows the output and multiplies the torque, useful for pulling power.

References

A quick note on where the physics comes from. The gear ratio as a tooth-count ratio, and the speed-for-torque trade that keeps power constant, are standard mechanics, set out in OpenStax's University Physics and described in the Wikipedia article on gear trains. The newton-metre and the other units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 10.7, Newton's Second Law for Rotation (torque and rotational power). https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation
2. Wikipedia, Gear train (gear ratio and torque). https://en.wikipedia.org/wiki/Gear_train
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811