Rectangular to Polar Coordinates Calculator

Created by Gabriela Diaz
Reviewed by Anna Szczepanek, PhD
Last updated: Sep 14, 2022

Welcome to Omni's rectangular to polar coordinates calculator, the tool that makes converting rectangular to polar coordinates an easy task!

If you've ever wondered what rectangular coordinates are or how they differ from polar coordinates, this is your place. Here, we'll also learn how to convert rectangular coordinates to polar coordinates with some simple trigonometry and the Pythagorean theorem.

Let's get started! 🏁

What are rectangular coordinates?

If we want to indicate the location of a point in space, we can do so through the so-called rectangular coordinate system.

This system, also known as the Cartesian coordinate system in two dimensions, allows us to easily represent the position of any point as an ordered pair of values (x, y), where x is the horizontal coordinate associated with the x-axis and y is the vertical coordinate linked to the y-axis.

What are polar coordinates?

The polar coordinate system is another useful coordinate system we can use to represent a point's position. By specifying the radius (or distance from a reference point) and the angle of rotation, we can describe the location of a point.

Often we use the letter r to denote the radius and the Greek letter θ (theta) to represent the angle. Similar to the rectangular coordinate system, in the polar form, we express the position of a point as an ordered pair (r, θ).

At this point, you might be wondering: Is the rectangular coordinate system better than the polar? When to use one or the other is mainly determined by which notation is more convenient for the type of problem you have to solve. Usually, when working with circles, cylinders, rotational motion, or periodic problems, we'll find the polar coordinates easier to apply. In contrast, the rectangular form could complicate the math of these problems.

How do I convert rectangular coordinates to polar coordinates?

To convert from the rectangular to the polar form, we use the following rectangular coordinates to polar coordinates formulas:

r = √(x² + y²)

θ = arctan(y / x)

Where:

  • x and y — Rectangular coordinates;
  • r — Radius of the polar coordinate; and
  • θ — Angle of the polar coordinate, usually in radians or degrees.

With these results, we express the polar coordinate as: (r, θ).

Notice that we can express a specific rectangular point as more than one polar point. For example, the rectangular coordinate (1, √3) can be written in the polar form as (2, π/3), or (2, 7π/3), or (2, 13π/3), or (2, π/3 + 2πn), or even (2, -5π/3). Here π represents the angular measure in radians of half a circle (or 180°) and n the number of cycles.

In this case, we have a unique value for the radius (r = 2) and a wide number of possible angles that can suit and perfectly represent the rectangular point (1, √3).

Using the rectangular to polar coordinates calculator

The rectangular to polar coordinates calculator couldn't make converting from rectangular to polar coordinates easier. By just entering the x and y values of your problem, the calculator will instantly display the r and θ components of the polar coordinate representation of your point. Yes, it's that simple! 😎

More coordinates conversion tools

If you enjoyed using this rectangular to polar coordinates calculator, then you might like to take a look at some of our other coordinates conversion tools:

FAQ

What's the difference between rectangular and polar coordinates?

We use rectangular coordinates to represent any point in a plane as the ordered pair (x, y), whereas, with polar coordinates, we can locate this same point by indicating its radius r from the origin and the counterclockwise angle from the x-axis θ, as (r, θ).

Are rectangular and cartesian coordinates the same?

Yes. In two dimensions, the cartesian coordinates are also known as rectangular coordinates. This type of coordinate notation allows us to represent any point in a plane as a pair of elements (x, y).

How do I convert the rectangular coordinate (3, 4) to polar coordinates?

To convert the rectangular coordinate (3, 4) into polar coordinates:

  1. To calculate the radius r, we use:

    r = √(x² + y²)

  2. Substitute the values for x = 3 and y = 4 and perform the corresponding operations:

    r = √(3² + 4²)
    r = 5

  3. Calculate the angle θ = arctan(y / x), by substituting the x and y values:

    θ = arctan(4 / 3)
    θ = 53.13°

  4. Finally, indicate the polar coordinate as a pair:

    (r, θ) = (5, 53.13°)

Gabriela Diaz
Rectangular (x, y) to Polar (r, θ)
x
y
r
θ
deg
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