Polar to Cartesian Converter
Two Ways to Name the Same Point
Every point on a plane can be located two different ways: Cartesian coordinates give its horizontal and vertical distance from the origin, while polar coordinates give its straight-line distance and the angle from a reference direction. Both describe the exact same point — converting between them is a matter of trigonometry, not measurement.
The Formulas
Polar to Cartesian: x = r×cos(θ), y = r×sin(θ)
Cartesian to Polar: r = √(x²+y²), θ = atan2(y, x)
Cartesian to Polar: r = √(x²+y²), θ = atan2(y, x)
Worked Examples
| Direction | Input | Output |
|---|---|---|
| Polar → Cartesian | r = 10, θ = 30° | x = 8.6603, y = 5 |
| Cartesian → Polar | x = 3, y = 4 | r = 5, θ = 53.1301° |
The second example is the familiar 3-4-5 right triangle, so r comes out to a clean 5.
Where This Matters
- Radar and sonar tracking — targets are naturally located by range and bearing (polar), then converted to x/y for mapping and plotting.
- Robotics arm control — a robotic arm often reasons in polar terms (reach and angle) while the workspace is defined in Cartesian coordinates.
- Complex numbers and signal processing — polar form (magnitude and phase) and rectangular form (real and imaginary parts) are the same conversion problem in a different notation.
How to Use This Calculator
- Choose the direction: Polar to Cartesian or Cartesian to Polar.
- For polar to Cartesian, enter Radius (r) and Angle (theta), then choose Degrees or Radians.
- For Cartesian to polar, enter x and y, and choose whether the angle result should be in Degrees or Radians.
- Select Calculate to get the converted coordinates.
Related Calculations
See these same cosine and sine relationships applied at radius 1 in the Unit Circle Calculator.