Spherical Coordinates Calculator – Guide & Formulas
Convert between Cartesian (x, y, z) and spherical (ρ, θ, φ) coordinates instantly. Supports 3D point conversion with step-by-step solutions.
Convert between Cartesian coordinates (x, y, z) and spherical coordinates (ρ, θ, φ) for any 3D point. Get the conversion formulas and step-by-step solutions.
Key Takeaway
Use the free Spherical Coordinates Calculator to convert between cartesian (x, y, z) and spherical (ρ, θ, φ) coordinates instantly. supports 3d point conversion with step-by-step solutions. Get instant results with step-by-step explanations.
How to Use the Spherical Coordinates Calculator
- Select the conversion direction: Cartesian to Spherical or Spherical to Cartesian.
- Enter the coordinates for your chosen system.
- Click "Calculate" to see the converted coordinates.
- Review the converted values and the conversion formulas used.
The Formula
Variable Definitions
- ρ (rho): Radial distance from origin to point
- θ (theta): Azimuthal angle in the xy-plane from x-axis (0 to 2π)
- φ (phi): Polar angle from the positive z-axis (0 to π)
- x, y, z: Cartesian coordinates
Example: Convert (1, 1, 1) to Spherical Coordinates
Convert the Cartesian point (1, 1, 1) to spherical coordinates.
- Step 1: Calculate ρ = √(1² + 1² + 1²) = √3 ≈ 1.732.
- Step 2: Calculate θ = arctan(y/x) = arctan(1/1) = 45° = π/4.
- Step 3: Calculate φ = arccos(z/ρ) = arccos(1/√3) ≈ 54.74°.
- Step 4: Spherical coordinates: (ρ, θ, φ) = (√3, 45°, 54.74°).
- Step 5: Verify: x = √3 sin(54.74°) cos(45°) ≈ 1 ✓
Frequently Asked Questions
What are spherical coordinates?
Spherical coordinates represent a point in 3D space using three values: distance from origin (ρ), azimuthal angle (θ), and polar angle (φ).
How do I convert Cartesian to spherical?
Use ρ = √(x² + y² + z²), θ = arctan(y/x), and φ = arccos(z/ρ).
How do I convert spherical to Cartesian?
Use x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), and z = ρ cos(φ).
What is the difference between θ and φ?
θ (theta) is the angle in the xy-plane from the x-axis (like longitude). φ (phi) is the angle from the z-axis (like colatitude). Conventions vary by field.
What is ρ in spherical coordinates?
ρ (rho) is the radial distance from the origin to the point. It is always non-negative and equals √(x² + y² + z²).
When should I use spherical coordinates?
Spherical coordinates are useful for problems with spherical symmetry, such as gravitational fields, electromagnetic radiation, and 3D graphics.
What are the ranges of the angles?
Typically θ ranges from 0 to 2π (360°) and φ ranges from 0 to π (180°). This covers all points in 3D space.
Is there more than one convention?
Yes, physics and mathematics sometimes swap θ and φ. This calculator uses the ISO convention: θ for azimuth, φ for polar angle.
How do I find the volume element?
In spherical coordinates, dV = ρ² sin(φ) dρ dθ dφ. This is essential for triple integrals in spherical coordinates.
Can I use this for cylindrical coordinates?
No, cylindrical coordinates (r, θ, z) are different. However, you can convert between spherical and cylindrical using r = ρ sin(φ) and z = ρ cos(φ).