Polar to Rectangular Coordinates Calculator

Created by Anna Szczepanek, PhD
Reviewed by Davide Borchia
Last updated: Feb 02, 2023

Converting polar coordinates to rectangular coordinates can be tedious, so here we are with this polar to rectangular coordinates calculator to save your time and effort. Never again will you wonder how to find rectangular coordinates from polar coordinates!

Polar and rectangular coordinates

Let's recall that the polar and rectangular coordinates are two different systems in math that serve the same purpose: they allow us to identify a point on a plane. They differ in how they look at a given point:

The polar system sees the world in terms of circles: to describe a point, imagine a circle centered at the plane's origin and such that it goes through our point. Draw the radius that touches the point: the length of this radius (the r coordinate) and the angle between this radius and the horizontal axis (the φ coordinate) are all we need to identify a point.

The rectangular system, in turn, sees the world in terms of horizontal and vertical directions. To identify a point, we note its position with regard to the vertical axis (the x coordinate) and to the horizontal axis (the y coordinate).

How do I convert polar coordinates to rectangular coordinates?

To go from the polar coordinates to the rectangular coordinates, i.e., form (r,φ) to (x,y), apply the following conversion formula:

x = r * cos φ
y = r * sin φ

Polar and rectangular coordinates.

As you can see, converting polar coordinates to rectangular coordinates is not hard in theory, yet may be hard in practice since you have to determine the sine and cosine of φ. Omni's polar to rectangular coordinates calculator is here to help you with that!

How to use this polar to rectangular coordinates calculator?

To find rectangular coordinates from polar coordinates using our tool, input the polar coordinates and watch how the Omni magic happens - the rectangular coordinates are already there! You only need to make sure the coordinate r you've entered is non-negative.

Omni coordinate resources

Satisfied with our polar to rectangular coordinates calculator? Want to learn more about various coordinate systems used in maths? Here's a few links you may want to explore:

FAQ

Can all polar coordinates be written as rectangular coordinates?

Yes, every point (r,φ) in the polar coordinates can be converted to rectangular coordinates (x,y), provided that r is non-negative.

How do I convert polar coordinates (5, π) to rectangular coordinates?

To find rectangular coordinates:

  1. Compute the sine and cosine of the angle we're given: sin π =0 and cos π = -1.
  2. Determine x as x = r * cos φ = -5.
  3. Determine y as x = r * sin φ = 0.
  4. Gathering the results, we can easily write down our point in rectangular coordinates: (-5,0).
Anna Szczepanek, PhD
Polar (r, θ) to rectangular (x, y)
r
θ
deg
x
y
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