Cartesian (x,y) to Polar (r, θ)
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y
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θ
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Polar (r, θ) to Cartesian (x,y)
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# Polar Coordinates Calculator

By Bogna Haponiuk

This polar coordinates calculator is a handy tool that allows you to convert Cartesian to polar coordinates, as well as the other way around. It is useful only in a 2D space - for 3D coordinates, you might want to head to our cylindrical coordinates calculator. This article will provide you with a short explanation of both types of coordinates and formulas for quick conversion.

## Cartesian and polar coordinates As you probably know, coordinates are used to uniquely describe the position of a point in space. For know, we will limit ourselves to a 2D space. It means that we only have two dimensions: height and width (no depth), just as on a piece of paper.

The Cartesian coordinate system is created by drawing two lines perpendicular to each other. Then, the point where they meet is called the origin of the coordinate system. Coordinates of any arbitrary point in space are the distances between this point and the two lines, denoted the x-axis and the y-axis.

The polar coordinate system, on the other hand, does not include any perpendicular lines. The origin of the polar system is a point, called the pole. An arbitrary ray from this point is chosen to be the polar axis. To find the polar coordinates of a given point, you first have to draw a line joining it with the pole. Then, the point's coordinates are the length of this line r and the angle θ it makes with the polar axis.

Our polar coordinates calculator is able to convert between Cartesian and polar coordinates.

## Converting from Cartesian to polar

Let's assume you know the Cartesian coordinates of a point, but want to express them as polar coordinates. (Our Cartesian to polar calculator assumes that the origin of the Cartesian system overlaps with the pole of the polar system). You have to use the following formulas for conversion:

`r = √(x² + y²)`

`θ = arctan (y/x)`

where

• (x, y) are the Cartesian coordinates;
• (r, θ) are the polar coordinates.

The polar coordinates are subject to the following constraints:

• r must be equal or greater than 0;
• θ has to lie within the range (−π, π].

## Converting from polar to Cartesian

It is also possible that you know the polar coordinates of a point, but wish to find the Cartesian ones with our polar coordinates calculator. To do it, simply use the following equations:

`x = r * cos θ`

`y = r * sin θ`

You can notice that the value `y/x` is the slope of the line joining the pole and the arbitrary point.

Bogna Haponiuk