# Gear Ratio Calculator

Created by Davide Borchia
Last updated: Jul 12, 2022

Kick in high gear with our gear ratio calculator: here, you will learn how gear multiplies speed and torque using only a couple of simple mathematical formulas. Keep reading our article to find out:

• What is a gear?
• What is the gear ratio?
• How to find the gear ratio of a gear train;
• How to use the gear ratio to calculate the speed of the output gear; and
• The formula that relates the gear ratio and the torque.

## What is a gear? And how do gears work?

A gear is one of the fundamental elements of every machine. Gears are everywhere: your car is full of them (and if you don't have a car, your bike literally is gears).

But what is a gear? A gear is a toothed wheel, a rotating device covered with cogs.

Each cog is a tooth of the gear. When you mesh two gears, you obtain a mechanism composed of two cogwheels (the name changes, but don't worry: we will keep calling them gears). If you connect to gears with a chain (as in a bike), then you get two sprockets.

There are, of course, many types of gears. They can differentiate:

• In the shape;
• In the shape of the cogs (helical, spur);
• In the orientation of the cogs: they can be vertical or horizontal, inside or outside;
• In the direction and type of the transferred motion (rotation, linear; clockwise, anticlockwise);

And so on.

The classic gear we all know is, most likely, an involute gear of the type called a spur.

When you pair (mesh) two gears, some engineering magic happens: the torque and speed are transmitted from one gear to the other. What we just build is a gear train, and here it is:

Let's take a better look at that gear train: the most immediate thing is that turning one gear causes the other to turn too. The gear which causes the rotation is called the driving gear, while the one "receiving" it is, of course, called the driven gear.

In a more complex gear train, we can identify many pairs of driving-driven gears and an input and one or more output gears. An exemplary gear train is represented in the pictures below:

## How to calculate the gear ratio?

The gear ratio is a quantity defined for each couple of gears: we calculate the gear ratio as the ratio between the circumference of the driving gear to the circumference of the driven gear:

$i=\frac{\pi\cdot d_{\text{driving}}}{\pi \cdot d_{\text{driven}}}$

Where $d_i$ is the diameter of the $i^{\text{th}}$ gear. As you can clearly see, we can simplify this equation in two ways. As the ratio of the diameters:

$i=\frac{d_{\text{driving}}}{d_{\text{driven}}}$

And as the ratio of the radii:

$i=\frac{r_{\text{driving}}}{r_{\text{driven}}}$

These versions of the gear ratio formula require you to know the size of the gears, or to measure them. What if we told you that you could also learn how to calculate the gear ratio by counting the number of teeth?

Measure the thickness of the gear's teeth, and their spacing. We can then write the gear ratio formula:

$i=\frac{n_{\text{driving}}\cdot (t_{\text{thickness}}+t_{\text{spacing}})}{n_{\text{driven}}\cdot (t_{\text{thickness}}+t_{\text{spacing}})}$

This is pretty much equivalent to calculating the gear ratio with the diameter of the gears; however, there's a lucky catch! The teeth's parameters must be the same: this allows to finally simplify the gear ratio formula to:

$i=\frac{n_{\text{driving}}}{n_{\text{driven}}}$

If you want, substitute driven with output and driving with input to use the alternative nomenclature.

## How to express the gear ratio?

Now you know how to find the gear ratio: but how do we express it? For now, you've seen the decimal form; however, we can express it in three different ways:

• As a decimal number: this way, we immediately know how many turns of one gear correspond to a turn of the other.
• As a fraction: this is particularly useful in subsequent calculations.
• As a ratio: two integer numbers separated by a colon. The form $\text{a}:\text{b}$ tells us that one gear needs $\text{a}$ turns and the other $\text{b}$ turns to return to the original position.

## Applications of the gear ratio formula: using the gear ratio to calculate speed and torque

The gear ratio gives us indications of the rotational speed of the elements of a gear train. However, we often need to know also the effective advantage we'd get by using that particular combination of gears.

In this section, we will introduce and teach you how to calculate the mechanical advantage of a gear train.

Calculating the mechanical advantage is straightforward: we simply have to take the inverse of the reciprocal of the gear ratio. This means that we don't really need to calculate it to find out how to apply it: we can simply use the gear ratio formula!

We use the gear ratio to calculate the speed of the output gear given the speed of the input one. How?

$\omega_{\text{out}} = i\cdot \omega_{\text{in}} = \frac{n_{\text{in}}}{n_{\text{out}}} \cdot \omega_{\text{in}}$

Notice that we are calculating the angular velocity: we can use the gear ratio to calculate the RPM, rotation per minute: the tangential speed is necessarily the same: if this was not true, the gears would slip.

Take this example: an input gear with $n_{\text{in}}=14$ meshing with an output gear with $n_{\text{in}}=14$.

Let's say the input gear is turning at a speed of $30\ \text{RPM}$. Can you use the gear ratio to calculate the RPM of the output gear too?

$\omega_{\text{out}}= \frac{42}{14}\cdot 30\ \text{RPM}=10\ \text{RPM}$

As you can see, the bigger gear spins slowlier than the input one.

We can't use the gear ratio to calculate the torque in the same way: in fact, this quantity decreases with an increase in speed. We need to use the gears' mechanical advantage:

$T_{\text{out}}=\frac{1}{i}\cdot T_{\text{in}} = \frac{n_\text{out}}{n_\text{in}}\cdot T_{\text{in}}$

Back to the example. Let's say that you are feeding the smaller gear with a torque of $20\ \text{N}\cdot\text{m}$. What's the torque on the bigger gear?

$T_{\text{out}} = \frac{42}{14} \cdot 20\ \text{N}\cdot\text{m} = 60\ \text{N}\cdot\text{m}.$

Thanks to the gear ratio, the torque increased!

## Connecting more than one gear: the gear ratio formula for complex gear trains

If your gear train is made of more than one gear, what will happen to the speed of the output gear?

Let's consider the following gear train: What will be the effect of the third gear on the rotation of the gear train?

We are considering the input to be the bigger gear on the left. We can define two gear ratios:

$i_{1-2} =\frac{n_1}{n_2}= \frac{32}{12}$

For the first pair, and:

$i_{2-3} = \frac{n_2}{n_3} =\frac{12}{20}$

For the second pair. Say that the first gear is rotating at $60\ \text{RPM}$. Calculate the speed of the last gear:

\begin{align*} v_2 &= i_{1-2}\cdot v_1 = \frac{32}{12}\cdot 60\ \text{RPM}\\ \\ & = 160\ \text{RPM} \end{align*}

Now calculate the speed for the last gear:

\begin{align*} v_3 &= i_{2-3}\cdot v_2= \frac{12}{20}\cdot 160\ \text{RPM}\\ \\ & = 96\ \text{RPM} \end{align*}

Now, just for "fun", try to pass directly from the first to the last gear:

\begin{align*} v_3 &=i_{1-3}\cdot v_1 = \frac{32}{20}\cdot 60\ \text{RPM}\\ \\ & = 96\ \text{RPM} \end{align*}

Same result! This is because gear ratios are multiplicative:

$i_{1-3}\!=\!i_{1-2}\!\cdot\! i_{2-3}\!=\! \frac{32}{12}\!\cdot\!\frac{12}{20}=\frac{32}{20}$

Do you want an easier explanation: the tangential velocity of the gears is conserved! The circumference of the middle gear moves as fast as the one of the bigger one: for the last gear, there's no difference... apart from the direction of the rotation. What we did is to insert an idle gear, which maintains the speed but inverts the direction of the output gear!

Davide Borchia
Input gear teeth number
Output gear teeth number
Gear ratio
:1
Gear ratio effects
Input rotational speed
rpm
Output rotational speed
rpm
Input torque
Nm
Output torque
Nm
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