Polar Coordinates Calculator

Created by Davide Borchia
Last updated: Jun 30, 2022

You don't need to meet bears or penguins to learn about polar coordinates: our polar coordinates calculator will introduce you to this alternative coordinate system. Here you will learn:

• What are polar coordinates;
• When do we use them;
• How to convert from rectangular coordinates to polar coordinates;
• How to convert from polar coordinates to Cartesian coordinates;

What are polar coordinates?

A sheet of paper is naturally described by the Cartesian coordinates system (also known as rectangular). As we humans tend to think in terms of left and right, forward and backward, and eventually up and down, a set of orthogonal axes is often well-fitting to describe our world. In two dimensions, we measure the distance of a point from the horizontal and vertical axes, and we call those quantities $x$ and $y$.

However, there are situations where it is convenient to think in terms of other quantities. This is the case of rotational motions. In this case, the angle and the distance from a "center" serve better the purpose of defining a point in a plane. Think of physical quantities as the centrifugal force, trajectories like Earth's orbit, but also objects with some degree of radial symmetry as .

In the polar coordinate system, each point of a plane is univocally determined by:

• The distance from a reference point; and
• The angle between a reference direction and the segment between reference and point.

Mathematicians call the reference point pole* and the reference direction polar axis. The two coordinates are called, respectively:

• Radius, or radial distance, $r$; and
• Azimuth, or polar angle, $\theta$.

🙋 Pendulums are perfectly described by polar coordinates: learn more about these surprisingly simple but complex objects at our simple pendulum calculator and physical pendulum calculator.

There are some constraints on the coordinates, in contrast to Cartesian coordinates:

• The radius can't be negative: $r\geq t$;
• The azimuth is defined in a round angle: $0\text{\textless}\theta\leq360\degree$.

How to convert from Cartesian to polar coordinates

Take two Cartesian coordinates, $x$, and $y$. To calculate the polar coordinates corresponding to that pair, we need to use a bit of trigonometry, but don't worry: it's basic one.

First, let's calculate the polar coordinates' radius:

$\footnotesize r = \sqrt{x^2+y^2}$

The first step in the calculations from Cartesian to polar coordinates is straightforward: it corresponds to the calculation of the hypotenuse of a right triangle.

To complete the conversion to polar coordinates, we need to calculate the azimuth. To do so, we can use the function $\operatorname{atan2}$, a slightly modified version of the arctangent function:

$\theta = \operatorname{atan2}(x,y)$

The function $\operatorname{atan2}$ is defined as:

$\footnotesize \operatorname{atan2}(x,y)= \begin{cases} \arctan(\frac{y}{x})&\text{if}\ x\text{\textgreater}0\\ \\ \arctan(\frac{y}{x})+\pi&\text{if}\ x\text{\textless}0,\ y\geq0\\ \\ \arctan(\frac{y}{x})-\pi&\text{if}\ x\text{\textless}0,\ y\text{\textless}0\\ \\ \frac{\pi}{2}&\text{if}\ x=0,\ y\text{\textgreater}0\\ \\ -\frac{\pi}{2}&\text{if}\ x=0,\ y\text{\textless}0\\ \\ \text{undefined}&\text{if}\ x=0,\ y=0 \end{cases}$

If you already know the radius, you can use an alternative (and slightly easier) function to calculate the polar coordinates' azimuth:

$\footnotesize \theta= \begin{cases} \arccos{(\frac{x}{r})}&\text{if}y\geq 0, r\neq0\\ \\ -\arccos{(\frac{x}{r})}&\text{if}y\text{\textless}0\\ \\ \text{undefined}&\text{if}r=0 \end{cases}$

Now that you know how to find the polar coordinates from the Cartesian ones, let's calculate the other way round.

Polar to cartesian coordinates calculator

We will now see the equations to calculate the polar to cartesian conversion.

The conversion from the angle-radius coordinates to $x$ and $y$ is fairly simple. Using basic trigonometry, we can calculate:

\begin{align*} x&=r\cdot \cos{\theta}\\ y& = r\cdot \sin{\theta} \end{align*}

From the equations of the polar to Cartesian calculations you can easily see the close relationship between polar coordinates and trigonometry.

Now you know how to convert polar coordinates both from and to Cartesian coordinates: our calculator will do it in no time, but it's better to know the math before the calculations!

Davide Borchia
Cartesian (x, y) to Polar (r, θ)
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Polar (r, θ) to Cartesian (x, y)
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