Last updated October 10, 2026 · Report an error
A point in 3D space can be described by Cartesian coordinates (x, y, z) or spherical coordinates (r, θ, φ — distance from the origin, inclination from the +z axis, and azimuth around the xy-plane). This tool converts a point between the two forms in either direction.
How it works
Choose a conversion direction. Cartesian → Spherical: enter x, y, and z, and the tool computes r = √(x² + y² + z²), θ = arccos(z/r), and φ = atan2(y, x). Spherical → Cartesian: enter r, θ, and φ in degrees, and the tool computes x, y, and z from them. This tool uses the physics/ISO convention, where θ is measured from the +z axis (not from the xy-plane, as some math textbooks define it).
- Enter conversion direction.
- Enter x (for Cartesian → Spherical).
- Enter y (for Cartesian → Spherical).
- Enter z (for Cartesian → Spherical).
- Enter r (for Spherical → Cartesian).
- Enter θ in degrees, inclination from +z axis (for Spherical → Cartesian).
- Enter φ in degrees, azimuth in xy-plane (for Spherical → Cartesian).
- Click Calculate to see your results.
Examples
A symmetric point
The point (1, 1, 1) converts to r ≈ 1.732, θ ≈ 54.74°, φ = 45° in spherical coordinates — and converting back reproduces (1, 1, 1) exactly.
Who should use it
- Physics and engineering coursework involving 3D coordinate systems.
- 3D graphics, geodesy, and antenna/radiation-pattern calculations.
Industry applications
- Physics and engineering education
- 3D graphics, robotics, and geodesy
Advantages
- Instant, exact conversion in either direction.
- Clearly documents which angle convention is used, avoiding a common source of confusion.
Limitations
- 3D spherical coordinate conventions genuinely differ across textbooks — always double check which one a downstream calculation expects.
Common mistakes to avoid
- Confusing the physics convention (θ from the +z axis) with the math-textbook convention some sources use (θ from the xy-plane) — they're not interchangeable without adjusting the formula.
- Entering θ outside the valid 0°-180° range — inclination angles don't wrap the way azimuth angles do.
Best practices
- Confirm which θ/φ convention your downstream formula or textbook expects before mixing results from different sources.
- Use the origin guard as a hint — if you get a validation error, your point genuinely has no defined direction.
Tips
- Converting a point to spherical and back to Cartesian should return your original x, y, and z — a handy way to sanity-check your own by-hand math.