Convert a point in the Cartesian plane to its equal polar coordinates with this polar coordinate calculator. Polar coordinates also take place in the x-y plane but are represented by a radius and angle, as shown in the diagram below.

Polar Coordinates Formula

The following formulas are used to convert polar coordinates from Cartesian coordinates.

r = √(x² + y²)
θ = arctan (y/x)
  • Where r is the radius
  • x and y are the coordinate points
  • θ is the angle

Polar Coordinates Definition

Polar coordinates are a system used to represent points in a two-dimensional plane. Unlike the commonly used Cartesian coordinates (x, y), polar coordinates express a point’s position using two values: the distance from a fixed point called the origin, and the angle between a reference direction, usually the positive x-axis, and a line connecting the origin to the point.

In polar coordinates, a point is represented by (r, θ), where r is the distance from the origin to the point and θ is the angle measured in radians. Radians are a unit of angular measure that offers a more natural and mathematically convenient way to work with angles compared to degrees.

The importance of polar coordinates lies in their ability to simplify certain mathematical calculations and describe certain phenomena more elegantly. They are particularly useful in fields such as physics, engineering, and mathematics, where circular or rotational symmetry is present. Polar coordinates make it easier to analyze and understand such symmetrical patterns.

How to calculate polar coordinates

As you can see from the formulas, the radius is a function of the square root of the sum of the x and y coordinates squared. In other words, the radius length acts as the hypotenuse on a triangle with lengths x and y.

The angle that describes the rotation of the radius is equal to an angle in a triangle with sides x and y, and hypotenuse r. This is also known as the reference angle.

More about Polar Coordinates

Polar coordinates have been around for a millennium. Coming about, as most math-related concepts are, through a Greek scientist. More specifically, a Greek astronomer was looking to use functions to calculate the position of stars.

The angle in polar coordinates is oftentimes described by the Greek letter theta and is measured in either degrees or radians. The radius denoted with r is typically measured in a unitless measure, but some distance measures such as inches or meters could be used.


What are polar coordinates?

Polar coordinates are a way of displaying the location of a point in the 2-dimensional plane using a radius of a circle and angle as measure from the x-axis.

What are Cartesian coordinates?

Cartesian coordinates are a way of display the location of a point in the 2 dimension plane using an X and Y coordinate.

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