Spherical Coordinates Calculator

Convert between Cartesian (x, y, z) and spherical (r, θ, φ) coordinates.

Choose Conversion Direction

Cartesian Coordinates

Result

Spherical Coordinates
(3.7417, 36.7°, 63.43°)
(r, θ, φ)
x1
y2
z3
r (radial distance)3.7417
θ (polar angle)36.7°
φ (azimuthal angle)63.43°
θ (radians)0.6405
φ (radians)1.1071

Step-by-Step Solution

r = sqrt(x² + y² + z²)

Understanding Spherical Coordinates

Spherical coordinates describe a point in three-dimensional space using three values: the radial distance r from the origin, the polar angle θ (theta) measured from the positive z-axis, and the azimuthal angle φ (phi) measured from the positive x-axis in the xy-plane.

Conversion Formulas

Cartesian to Spherical

Convert (x, y, z) to (r, θ, φ) using these formulas.

r = sqrt(x² + y² + z²)

Polar Angle θ

The angle from the z-axis down to the point.

θ = arccos(z / r)

Azimuthal Angle φ

The angle in the xy-plane from the x-axis.

φ = arctan2(y, x)

Spherical to Cartesian

Convert (r, θ, φ) back to (x, y, z).

x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ

Convention Note

This calculator uses the physics convention (ISO 31-11): θ is the polar angle (from z-axis, 0 to 180 degrees) and φ is the azimuthal angle (in the xy-plane, 0 to 360 degrees). Some mathematics textbooks swap θ and φ.

Applications

  • Physics: Gravitational and electric fields with spherical symmetry.
  • Astronomy: Celestial coordinates for star and planet positions.
  • Computer Graphics: Camera positioning, environment mapping.
  • Navigation: GPS coordinates are a form of spherical coordinates.