# Gear Ratio Calculator

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**:

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:

And as the **ratio of the radii**:

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:

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:

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?

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?

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:

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?

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:

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

For the first pair, and:

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

Now calculate the speed for the last gear:

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

Same result! This is because gear ratios are **multiplicative**:

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!