Polar Coordinates Calculator | Convert Polar & Cartesian (r,θ ↔ x,y)

What are Polar and Cartesian Coordinates?

Defining the two major 2D coordinate systems
Understanding their core components
Key differences and when to use each one

In mathematics, we use coordinate systems to uniquely determine the position of a point. The two most common systems for a two-dimensional plane are the Cartesian and Polar coordinate systems.

The Cartesian Coordinate System

The Cartesian system, named after René Descartes, describes a point using two perpendicular axes: the horizontal x-axis and the vertical y-axis. A point is identified by an ordered pair (x, y), representing its signed distances from the y-axis and x-axis, respectively. This rectangular grid is intuitive for representing linear relationships and geometric shapes like squares and rectangles.

The Polar Coordinate System

The Polar system describes a point's position using a distance and an angle. Each point is identified by an ordered pair (r, θ), where 'r' is the radial distance from a central point called the pole (or origin), and 'θ' (theta) is the angle measured counterclockwise from a fixed ray, the polar axis (usually aligned with the positive x-axis).

System Representation

Cartesian: The point (3, 4) is 3 units right and 4 units up.
Polar: The point (5, 53.13°) is 5 units away from the origin at an angle of 53.13 degrees.

Step-by-Step Guide to Using the Polar Coordinates Calculator

Selecting your conversion type
Entering your coordinate values correctly
Interpreting the calculated results accurately

This calculator simplifies the conversion process. Follow these steps for accurate results.

1. Choose Your Conversion Direction

Start by selecting your desired conversion from the 'Conversion Type' dropdown. Choose 'Polar to Cartesian' if you have (r, θ) values, or 'Cartesian to Polar' if you have (x, y) values.

2. Select the Angle Unit (for Polar Inputs)

If converting from Polar to Cartesian, you must specify whether your angle (θ) is in 'Degrees' or 'Radians'. The calculator will also return the angle in this selected unit when converting from Cartesian to Polar.

3. Input Your Coordinates

Fill in the corresponding input fields based on your chosen conversion type. Ensure the radius 'r' is a non-negative number.

4. Calculate and View Results

Click the 'Calculate' button. The results will appear in the 'Result' card, showing the converted coordinates. You can easily copy the results using the copy button.

Usage Scenarios

Inputting r=10, θ=90, unit=degrees for Polar to Cartesian.
Inputting x=-5, y=5 for Cartesian to Polar.

Mathematical Derivation and Formulas

The trigonometric foundations of conversion
Formulas for Polar to Cartesian
Formulas for Cartesian to Polar

The conversion between polar and Cartesian coordinates is based on right-triangle trigonometry.

From Polar (r, θ) to Cartesian (x, y)

By forming a right triangle with the pole, the point (r, θ), and its projection on the x-axis, we can use basic trigonometric definitions (SOH CAH TOA). The hypotenuse is 'r', and the angle is 'θ'. The adjacent side is 'x' and the opposite side is 'y'.

x = r * cos(θ)

y = r * sin(θ)

From Cartesian (x, y) to Polar (r, θ)

To find 'r', we use the Pythagorean theorem on the same right triangle:

r = √(x² + y²)

To find the angle 'θ', we use the arctangent function. The atan2(y, x) function is preferred over atan(y/x) as it correctly handles all four quadrants.

θ = atan2(y, x)

Formula Application

For (r=2, θ=30°): x = 2 * cos(30°) = 1.732, y = 2 * sin(30°) = 1.
For (x=3, y=4): r = √(3² + 4²) = 5, θ = atan2(4, 3) = 53.13°.

Real-World Applications of Polar Coordinates

Applications in Physics and Engineering
Use in Computer Graphics and Robotics
Relevance in Navigation and Astronomy

Polar coordinates are not just an academic concept; they are essential in many practical fields.

Physics and Engineering

In physics, polar coordinates are ideal for describing objects moving in circles, such as planets in orbit or a car on a circular track. They are also fundamental in analyzing fields that have a central source, like electromagnetic fields or gravitational fields.

Engineers use them to describe the radiation patterns of antennas and microphones, where signal strength varies with angle.

Computer Graphics and Robotics

In computer graphics, rotating an object around a pivot point is much simpler using polar coordinates. They are also used to create circular or spiral patterns. In robotics, a robotic arm's position is often defined by an angle and an extension length, a direct application of polar coordinates.

Practical Fields

A radar screen showing an aircraft's position by its distance and bearing (angle).
A CAD program rotating a component by a specific angle.

Common Misconceptions and Correct Methods

The uniqueness of polar coordinates
Handling negative angles and radii
Choosing the right arctangent function

While powerful, polar coordinates have nuances that can lead to confusion.

Are Polar Coordinates Unique?

Unlike Cartesian coordinates, the polar representation of a point is not unique. For example, the point (r, θ) can also be represented as (r, θ + 360°) or (r, θ - 360°). The pole (origin) itself is (0, θ) for any angle θ. This calculator provides the principal value for the angle, typically in the range [0, 360°) or [0, 2π).

The Importance of atan2(y, x)

A common mistake is to calculate θ using atan(y/x). This function only returns values between -90° and +90°, failing to distinguish between diagonally opposite quadrants (e.g., I and III). The atan2(y, x) function considers the signs of both x and y to return the correct angle in the full 360° range, which is why it's essential for accurate conversions.

Avoiding Pitfalls

For (x=-1, y=-1), `atan(-1/-1)` = `atan(1)` = 45°, which is incorrect. `atan2(-1, -1)` = -135° or 225°, which is correct.
The point (5, 90°) is the same as (5, 450°).

Convert Polar to Cartesian

Convert Cartesian to Polar

Convert with Radians

Convert with Negative Coordinates