Cartesian to Spherical Converter
Convert 3D Cartesian coordinates (x, y, z) to spherical coordinates while choosing the math or physics convention and angle unit (degrees or radians).
Coordinate inputs
Units follow the selected angle setting.
How to use this converter
Enter the 3D Cartesian components (x, y, z), pick degrees or radians, and choose the math or physics convention. Click Calculate to update the radial distance and angles. The Reset button restores the default values so you can start fresh.
Methodology
The tool computes the radius using the Euclidean distance √(x² + y² + z²), then derives the azimuth with atan2(y, x) and the polar angle with arccos(z / ρ). The math and physics conventions simply swap which angle is labeled θ or φ. Angle units are converted after the raw radian values are calculated.
- The math convention treats θ as the azimuth in the x-y plane and φ as the polar angle from +z.
- The physics convention swaps the role of θ and φ to match common electromagnetics and quantum mechanics notation.
- Normalization keeps the azimuth within a positive cycle (0–360° or 0–2π rad) so quadrant ambiguity is eliminated.
- If the point is the origin (ρ = 0), angles default to 0°/rad for display convenience.
FAQ
What if ρ = 0?
If x = y = z = 0, the radius is zero and the angles technically lack direction. The converter returns 0 for both angles to keep the display consistent.
Why two different names (ρ vs r, θ vs φ)?
Different disciplines favor different labels. The math-friendly layout uses ρ for radius and φ for the polar angle, while physics notation prefers r and swaps θ/φ. This tool lets you match whichever reference you need.
Will it work with negative coordinates?
Yes. Using atan2 keeps the azimuth in the correct quadrant even when x or y are negative.
Full original guide (expanded)
Related math & conversions
About this tool
Suitable for multivariable calculus, electromagnetics, astronomy, and 3D geometry. Handles both main spherical systems and outputs in the angle unit you prefer so you can match lecture notes, textbooks, or simulation software.