Gear ratio calculator

What is a gear ratio calculator?

A gear ratio calculator works out how a pair of meshing gears trades rotational speed for turning force. Enter the number of teeth on the driving (input) gear and on the driven (output) gear, and the tool returns the gear ratio both as a decimal and in the familiar “ratio to one” form. Add an input speed or an input torque and it also reports the speed and torque delivered at the output shaft. It saves you from juggling the inverse relationships by hand and is handy for anyone designing a gearbox, tuning a bicycle drivetrain, or studying mechanics.

How gears trade speed for torque

When two gears mesh, their teeth move past the contact point at the same rate, so the gear with more teeth turns more slowly. A small driving gear spinning a large driven gear therefore produces a slower but stronger output: speed goes down and torque goes up by the same factor. Reverse the arrangement and you get the opposite, an overdrive that spins faster but pushes with less force. The gear ratio is simply the number that captures this trade. Idler gears placed between the input and output can reverse the direction of rotation, but because they mesh on both sides they do not change the overall ratio.

How does the calculator work?

The calculator first divides the driven gear’s tooth count by the driving gear’s tooth count to get the ratio. Because rotational speed is inversely proportional to tooth count, it divides the input speed by the ratio to find the output speed. Torque scales the other way, so it multiplies the input torque by the ratio to find the output torque. If the driving gear is left blank or set to zero, no ratio is defined and the results stay empty.

Formula

The gear ratio $i$ between a driving gear with $N_\text{in}$ teeth and a driven gear with $N_\text{out}$ teeth is:

$i = \frac{N_\text{out}}{N_\text{in}}$

The output rotational speed $\omega_\text{out}$ follows from the input speed $\omega_\text{in}$ as:

$\omega_\text{out} = \frac{\omega_\text{in}}{i} = \omega_\text{in}\,\frac{N_\text{in}}{N_\text{out}}$

The output torque $\tau_\text{out}$ follows from the input torque $\tau_\text{in}$ as:

$\tau_\text{out} = i\,\tau_\text{in} = \tau_\text{in}\,\frac{N_\text{out}}{N_\text{in}}$

Where:

• $N_\text{in}$ is the number of teeth on the driving (input) gear,
• $N_\text{out}$ is the number of teeth on the driven (output) gear,
• $i$ is the gear ratio,
• $\omega_\text{in}$ and $\omega_\text{out}$ are the input and output rotational speeds,
• $\tau_\text{in}$ and $\tau_\text{out}$ are the input and output torques.

A ratio greater than one is a reduction (slower, stronger output); a ratio less than one is an overdrive (faster, weaker output).

Examples

Example 1: A driving gear has $N_\text{in} = 10$ teeth and meshes with a driven gear of $N_\text{out} = 40$ teeth. Find the gear ratio.

1. Substitute the tooth counts into the formula: $i = \frac{N_\text{out}}{N_\text{in}} = \frac{40}{10}$

2. The result is $i = 4$, written as $4 : 1$. The input gear turns four times for each turn of the output gear.

Example 2: With the same $10 \to 40$ gear pair, the input shaft spins at $\omega_\text{in} = 1200$ rpm and delivers $\tau_\text{in} = 5$ N·m of torque. Find the output speed and torque.

1. The output speed divides by the ratio: $\omega_\text{out} = \frac{1200}{4} = 300 \ \text{rpm}$

2. The output torque multiplies by the ratio: $\tau_\text{out} = 4 \times 5 = 20 \ \text{N·m}$

3. The output shaft turns four times slower but delivers four times the torque.

Notes

• A ratio above $1$ reduces speed and increases torque; a ratio below $1$ does the reverse. A ratio of exactly $1 : 1$ passes speed and torque through unchanged.
• The speed and torque relationships assume an ideal, lossless gear pair. Real gears lose a few percent to friction, so the actual output torque is slightly lower.
• For a multi-stage gear train, multiply the individual stage ratios together to get the overall ratio.
• Because the ratio depends only on tooth counts, you can equally use the gears’ pitch diameters or radii, which are proportional to the number of teeth.

FAQs

Which gear is the “driving” gear?

The driving (input) gear is the one connected to the power source, such as a motor or pedal. The driven (output) gear is the one it turns, connected to the load.

What does a gear ratio of 4:1 mean?

It means the driving gear must complete four full turns for the driven gear to complete one. The output therefore turns four times slower and, in the ideal case, with four times the torque.

Does adding an idler gear change the ratio?

No. An idler gear placed between the input and output gears reverses the output’s direction of rotation but leaves the overall ratio unchanged, because it meshes with both gears.

How do I get more torque from a gear pair?

Use a larger driven gear or a smaller driving gear so that the ratio rises above one. The output then turns more slowly but with proportionally more torque.

Can the gear ratio be less than one?

Yes. When the driven gear has fewer teeth than the driving gear, the ratio falls below one. This is an overdrive: the output spins faster than the input but delivers less torque.

You can find this tool at https://www.mega-calculator.com/physics/gear-ratio/.