Input gear teeth number
Output gear teeth number
Gear ratio
Mechanical advantage
Gear ratio effects
Input rotational speed
Output rotational speed
Input torque
Output torque

This gear ratio calculator determines the mechanical advantage a two-gear setup produces in a machine. The gear ratio gives us an idea of how much an output gear is sped up or slowed down, or how much torque is lost or gained in a system. We equipped this calculator with the gear ratio equation and the gear reduction equation so you can quickly determine the gear ratio of your gears. To learn more about gear ratio calculation, and how it is essential in making simple machines (and even complicated ones), keep on reading.

But first: what is a gear?

A gear is a toothed wheel that can change the direction, torque and speed of rotational movement applied to it. Gears come in different shapes and sizes, and these differences describe the translation or transfer of the rotational movement. The transfer of movement happens when two or more gears in a system mesh together while in motion. We call this system of gears a gear train.

In a gear train, turning one gear also turns the other gears. The gear that initially receives the turning force, either from a powered motor or just by hand (or foot in the case of a bike), is called the input gear. We can also call it the driving gear since it initiates the movement of all the other gears in the gear train. The final gear that is influenced by the input gear is known as the output gear. In a two-gear system, we can call these gears the driving gear and the driven gear respectively.

The resulting movement of the output gear could be in the same direction as the input gear, but it could be in a different direction or axes of rotation depending on the type of gears in the gear train. To help you visualize this, here is an illustration of the different types of gears and their input-to-output gear relationships:

An illustration showing the different types of gears and how movements are translated between them

What is gear ratio and how to calculate gear ratio

The gear ratio is the ratio of the circumference of the input gear to the circumference of the output gear in a gear train. The gear ratio helps us in determining the number of teeth each gear needs to produce a desired output speed/angular velocity, or torque.

We calculate the gear ratio between two gears by dividing the circumference of the input gear by the circumference of the output gear. We can determine the circumference of a specific gear in the same way we calculate the circumference of a circle. In equation form, it looks like this:

gear ratio = (π * diameter of input gear)/(π * diameter of output gear)

Simplifying this equation, we can also obtain the gear ratio when just the gears' diameters or radii are considered:

  • gear ratio = (π * diameter of input gear)/(π * output gear)
  • gear ratio = (diameter of input gear)/(diameter of output gear)
  • gear ratio = (radius of input gear)/(radius of output gear)

Similarly, we can calculate the gear ratio by considering the number of teeth on the input and output gears. Doing so is similar to considering the circumferences of the gears. We can express the gear's circumference by multiplying the sum of a tooth's thickness and the spacing between teeth by the number of teeth the gear has:

gear ratio = (input gear teeth number * (gear thickness + teeth spacing)) / (output gear teeth number * (gear thickness + teeth spacing))

But, since the thickness and spacing of the gear train's teeth must be the same for the gears to engage smoothly, we can cancel out the gear thickness and teeth spacing multiplier in the above equation, leaving us with the equation below:

gear ratio = input gear teeth number / output gear teeth number

The gear ratio, just like any other ratios, can be expressed as:

  • a fraction or a quotient - where, if possible, we simplify the fraction by dividing both the numerator and the denominator by their greatest common factor.
  • a decimal number - expressing the gear ratio as a decimal number gives us a quick idea about how much the input gear has to be turned for the output gear to complete one full revolution.
  • an ordered pair of numbers separated by a colon, such as 2:5 or 1:14. With this, we can see the fewest number of turns required for both the input and output gears to return to their original positions at the same time.

From a different perspective, if we take the reciprocal of the gear ratio in its fractional form and simplify it to a decimal number, we get the value for the mechanical advantage (or disadvantage) our gear train or gear system has.

Understanding gear ratio and mechanical advantage values

Gear ratios are quite easy to understand, and now that we know how to calculate gear ratio, wouldn't it be better to know how it effects the gears themselves? To better explain gear ratios, let us consider a two gear system where the input and output gears have ten and forty teeth respectively:

A simple drawing of two gears with an 10-toothed input gear and a 50-toothed output gear

Following our gear ratio equation, we can say that this gear train has a gear ratio of 10:40, 10/40, or simply 1/4 (or 0.25). This gear ratio means that the output gear would only rotate 1/4 of a full rotation after the input gear has completed a full turn. Continuing in this fashion and keeping a consistent input speed, we see that the rate of the output gear is also 1/4 of that of the input speed. In other words, the speed of the input gear is four times the speed of the output gear, as can be seen in the animated image below:

An animation of a gear train with an input gear spinning four times faster than its output gear

While this setup demonstrates a gear reduction in terms of speed, in return it provides us with an output that has more torque, when compared to the input. The reciprocal of its gear ratio is 4/1, so we can say that we get four times the mechanical advantage when it comes to torque.

Important note on idler gears

A spur gear of any number of teeth between the input and output gears does not change the total gear ratio of the gear train. However, this gear (or gears) can change the direction of the output gear. We call this in-between gear as an idler gear. As an example, here is a 1:2.5 gear reduction system with an additional idler gear:

An animation of a gear train with idler gear

Without the idler gear, here is the same gear train. Note that the direction of the output gear is reversed:

An animation of a gear train without the idler gear

Real-life simple machines with gears

We see gears in our everyday lives, and to gain an even better understanding of gear ratios, here are some real-life examples of simple machines with gears in them:

Mechanical advantage in terms of speed

An image of a hand drill

Hand drills, though they seem to be less popular nowadays, are a great example of a simple machine that demonstrates a mechanical advantage in terms of speed. Cranking its handle will spin the drill bit at a high-speed.

Mechanical advantage in terms of torque

An image of a cyclist going uphill

Going uphill, riding a bike is easier if you are in a low-speed gear. Doing so results in better torque, providing more power when going uphill. This may mean we have to pedal more, but our ascend will be much easier. A bicycle sprocket-and-chain mechanism is much like a rack-and-pinion setup. The chain acts as a rack gear, directly transferring the motion to the rear bike sprocket.
Kenneth Alambra and Nicholas Swanson