1. What Are Spherical Coordinates?

Spherical coordinates represent points in 3D space using three values: radial distance (ρ), polar angle (θ), and azimuthal angle (φ). This system is particularly useful for problems with spherical symmetry.

2. Conversion Formulas

The calculator uses these conversion formulas:

Where:

• \( \rho \) — Radial distance from origin (always ≥ 0)
• \( \theta \) — Azimuthal angle in xy-plane from x-axis (-π to π radians)
• \( \phi \) — Polar angle from z-axis (0 to π radians)
• \( x, y, z \) — Cartesian coordinates

3. Applications

Details: Spherical coordinates are widely used in physics, engineering, and computer graphics, especially for systems with spherical symmetry like atomic orbitals, planetary motion, or 3D rendering.

4. Using the Calculator

Tips: Enter Cartesian coordinates (x, y, z) as real numbers. The calculator will output ρ (distance), θ (azimuthal angle), and φ (polar angle) in both radians and degrees.

5. Frequently Asked Questions (FAQ)

Q1: What's the range for θ and φ?
A: θ ranges from -π to π radians (-180° to 180°), φ ranges from 0 to π radians (0° to 180°).

Q2: How are negative x/y/z values handled?
A: The calculator correctly handles all real number inputs, with θ adjusting based on the quadrant.

Q3: What if ρ = 0 (origin point)?
A: At the origin (0,0,0), θ and φ are undefined but set to 0 for practical purposes.

Q4: How does this differ from cylindrical coordinates?
A: Cylindrical coordinates use (r, θ, z) where r is the radial distance in xy-plane only.

Q5: Can I convert back to Cartesian coordinates?
A: Yes, using: x = ρ sinφ cosθ, y = ρ sinφ sinθ, z = ρ cosφ.