Polar Coordinates Calculator
This polar coordinates calculator is a handy tool that allows you to convert Cartesian to polar coordinates, as well as the other way around. It is applicable only in a 2D space — for 3D coordinates, you might want to use a cylindrical coordinates calculator. Below you'll find a short explanation of both types of coordinates and formulas for quick conversion.
Cartesian and Polar Coordinates
We use coordinates to describe the position of a point in space uniquely. In a 2D space (a flat plane — like a piece of paper) we have two dimensions: horizontal and vertical.
The Cartesian coordinate system is created by drawing two perpendicular lines (axes). The point where they meet is called the origin. Coordinates of any point are the signed distances from this point to each axis, denoted x (horizontal) and y (vertical).
The polar coordinate system uses a fixed reference point called the pole and a ray from it called the polar axis. To locate a point you draw a line from the pole to the point. Its length is r and the angle it makes with the polar axis is θ.
Our calculator assumes the origin of the Cartesian system coincides with the pole of the polar system, and the polar axis points in the positive x direction.
How Do I Convert from Cartesian to Polar?
If you know the Cartesian coordinates (x, y) of a point and want to express them as polar coordinates (r, θ), use the following formulas:
- r = √(x² + y²)
- θ = arctan(y / x) (use atan2(y, x) to get the correct quadrant)
The polar coordinates are subject to the following constraints:
- r must be greater than or equal to 0
- θ lies within the range (−180°, 180°] or equivalently (−π, π] radians
How Do I Convert from Polar to Cartesian?
To go from the polar coordinates (r, θ) of a point to the Cartesian coordinates (x, y), use:
- x = r × cos(θ)
- y = r × sin(θ)
Note that y / x = tan(θ), which is the slope of the line joining the pole and the point.
Measurement Units
The calculator supports both metric (mm, cm, m, km) and US/Imperial (in, ft, yd, mi) distance units for the coordinate values and the radius r. You can also choose to work with dimensionless coordinates by selecting "no unit".
Angles can be expressed in degrees (°) or radians (rad). All results show both representations simultaneously for your convenience.
Quick Reference Table
| From | To | Formulas |
|---|---|---|
| Cartesian (x, y) | Polar (r, θ) | r = √(x² + y²) ; θ = atan2(y, x) |
| Polar (r, θ) | Cartesian (x, y) | x = r cos θ ; y = r sin θ |
Example: Cartesian (3, 4) → Polar
- r = √(3² + 4²) = √(9 + 16) = √25 = 5
- θ = arctan(4 / 3) ≈ 53.13° ≈ 0.9273 rad
Example: Polar (5, 53.13°) → Cartesian
- x = 5 × cos(53.13°) ≈ 3
- y = 5 × sin(53.13°) ≈ 4