Coordinate System Converter - Cartesian, Polar, Cylindrical & Spherical
Coordinate System Converter
Convert points between 2D and 3D coordinate systems
Five Systems
Start from any of 2D Cartesian, polar, 3D Cartesian, cylindrical or spherical and see every equivalent form.
Correct Quadrant
Angles use the two-argument arctangent, so (-1, -1) is -135°, not 45°.
Degrees or Radians
Type angles in either unit; results show both.
Private
Calculated in your browser.
The Conversion Formulas
In 2D, x = r cos θ and y = r sin θ; going back, r = √(x2 + y2) and θ = atan2(y, x). Cylindrical coordinates (ρ, φ, z) are polar coordinates in the xy-plane plus a height z. Spherical coordinates (r, θ, φ) give the distance from the origin r, the polar angle θ measured down from the positive z-axis, and the azimuth φ measured around from the positive x-axis: x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ.
This is the ISO 80000-2 convention used in physics and engineering. Many maths textbooks swap the names, calling the azimuth θ and the polar angle φ, so check which one your course uses before copying numbers. Example: the point (1, 1, 1) has spherical coordinates r = 1.732, θ = 54.7356° and φ = 45°.
Key Takeaways
- Physics convention: θ is measured from the z-axis and φ around it (ISO 80000-2).
- Angles in range: Azimuths are given from -180° to 180°; the polar angle from 0° to 180°.
- Round trips: Converting a result back gives the original point, apart from rounding.