Polar to Cartesian Coordinates Calculator

Created by Davide Borchia
Reviewed by Gabriela Diaz
Last updated: Aug 31, 2022

Our polar to cartesian coordinates calculator will make the conversion between these two coordinates systems as easy as they can: input the coordinates and let us do the math.

In this short article, you will learn:

  • What are polar and cartesian coordinates: similarity and differences between the two most common coordinate systems;
  • The elements of polar and cartesian coordinates;
  • How to convert polar coordinates to cartesian coordinates: calculations and explanation; and
  • When it's better to calculate the cartesian coordinates from the polar coordinates.

Coordinate systems: polar coordinates and cartesian coordinates

You've got a nice two-dimensional plane: how do you specify where a point lies? You have multiple choices, but it's likely that you will either use polar or cartesian coordinates.

Cartesian coordinates employ a set of orthogonal axis to span the plane. Every point of the plane is uniquely identified by a set of two numbers, the coordinates xx and yy.

Polar coordinates use two different quantities to identify a point on a two-dimensional plane:

  • A distance; and
  • An angle.

To use distance and angle to determine the position of a point, we need to choose a pole, corresponding to the origin of a Cartesian coordinate system, and a polar axis, that gives a reference on the distance of a point from the pole.

Once we set these objects, we can connect the point to the pole and measure:

  • ρ\rho or rr — The distance between point and pole; and
  • θ\theta or φ\varphi — The angle between the line connecting point and pole and the polar axis.

🙋 Scaling up the polar coordinates system to three dimensions gives us the spherical coordinates system. Learn more about it and how to convert from and to it with our spherical coordinates calculator!

How to convert polar coordinates to cartesian coordinates

To convert from polar coordinates to cartesian coordinates, you must take the two coordinates in the polar system (distance and angle) and apply the following set of rules:

x=rcos(θ)y=rsin(θ)\begin{align*} x &= r \cdot \cos{(\theta)}\\ y &= r \cdot \sin{(\theta)} \end{align*}

These equations come directly from the trigonometric circle, appropriately expanded — or shrunk — to cover the entire plane. Make the origin of a cartesian coordinate system and the pole of a polar one coincide together with the polar axis and xx axis. You can easily see that for points on the unit circle (circle with r=1r=1), the yy coordinate corresponds to the sine, while the xx coordinate corresponds to the cosine of the angle.

Other calculators to switch between coordinate systems

Depending on your needs, you may find yourself with the wrong coordinates system in hand. Omni is here to help with a set of useful tools to speed the conversion from one system to the others. Apart from our polar to cartesian coordinates calculator, you can find:

FAQ

How do I convert polar coordinates to cartesian coordinates?

To calculate the cartesian coordinates from the polar coordinates, make sure to know:

  • The distance from the point to pole r; and
  • The angle relative to the polar axis θ.

Then, to find the corresponding cartesian coordinates, apply the following equations:

  • x = r × cos(θ);
  • y = r × sin(θ).

When should I use cartesian coordinates instead of polar coordinates?

Use cartesian coordinates instead of polar coordinates:

  • When dealing with straight lines, polygons, and other non-curve objects;
  • When you need to perform translations (polar coordinates better handle rotations);
  • If you need a coordinate system with higher dimensionality than 3;
  • When your problem has higher symmetry.

What are the cartesian coordinates of (1,π/4)?

The cartesian coordinates of the point (1,π/4) are (√2/2,√2/2). The point lies on the unit circle, the first quadrant's bisectrix. To find the coordinates, apply the conversion from polar to cartesian system:

  • x = r × cos(θ) = 1 × cos(π/4) = √2/2; and
  • y = r × sin(θ) = 1 × sin(π/4) = √2/2.

Are polar coordinates unique?

Not precisely: since polar coordinates enjoy rotational symmetry, you can define a point with any angle equal to the original angle plus an integer multiple of , both positive and negative. In comparison, once you fix the axes, cartesian coordinates are unique.

As an example, take the point (1.37,π/3). You can't distinguish it from (1.37,7π/3), (1.37,13π/3), and so on.

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