1. What Are Spherical Coordinates?

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

2. How Does the Calculator Work?

The calculator uses these conversion formulas:

Where:

• \( r \) — Radial distance from origin
• \( \theta \) — Azimuthal angle in the xy-plane from x-axis
• \( \phi \) — Polar angle from z-axis
• \( x, y, z \) — Cartesian coordinates

Note: The calculator provides results in both radians and degrees for angles.

3. Applications of Spherical Coordinates

Details: Spherical coordinates are widely used in physics, engineering, astronomy, and computer graphics, especially when dealing with spherical symmetry or radial fields.

4. Using the Calculator

Tips: Enter the Cartesian coordinates (x, y, z) and the calculator will compute the corresponding spherical coordinates (r, θ, φ).

5. Frequently Asked Questions (FAQ)

Q1: What's the range for spherical coordinates?
A: Typically r ≥ 0, 0 ≤ θ < 2π, and 0 ≤ φ ≤ π.

Q2: How is this different from cylindrical coordinates?
A: Cylindrical coordinates use (r, θ, z) while spherical uses (r, θ, φ) for full 3D representation.

Q3: What's the right-hand rule in this context?
A: The coordinate system follows the right-hand rule where θ increases counterclockwise from the x-axis.

Q4: How are negative coordinates handled?
A: Negative Cartesian coordinates are valid and will produce appropriate angle values.

Q5: What about at the origin?
A: At (0,0,0), r=0 but angles are undefined (calculator will show 0).