Spherical Coordinates Calculator
Convert between rectangular (x, y, z) and spherical (r, θ, φ) coordinates. Pick a direction, then enter the three values to get r, the polar angle θ and the azimuth φ in degrees and radians (physics ISO convention).
Example: with Direction Rectangular → Spherical (x, y, z) · x (or r) 1 · y (or θ in degrees) 1 · z (or φ in degrees) 1 → Radial r / x: r = 1.7321.
- Polar θ / yθ = 54.7356° (0.9553 rad)
- Azimuth φ / zφ = 45° (0.7854 rad)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Physics/ISO convention: r is distance from the origin, θ is the polar angle down from the +z axis (0 to 180 degrees), and φ is the azimuth around the +z axis measured from +x. Many calculus texts swap the θ and φ labels.
Three numbers for a point in space
Spherical coordinates locate a point by how far it is and which way it faces. r is the straight-line distance from the origin. The polar angle θ measures how far the direction tips down from the +z axis, running from 0 at the north pole to 180 degrees at the south. The azimuth φ spins around the vertical axis, measured from the +x direction like a compass bearing.
Converting from rectangular is direct: r is the magnitude √(x²+y²+z²), θ is acos(z/r), and φ is atan2(y, x). Going the other way, x = r sinθ cosφ, y = r sinθ sinφ, and z = r cosθ. The only trouble spot is the origin, where every direction is equally valid and the angles are undefined.
Two conventions, one warning
Physics and engineering (the ISO 80000-2 standard) use θ for the polar angle and φ for the azimuth, which is what this tool does. Many US calculus textbooks reverse the letters, calling the azimuth θ and the polar angle φ, and often write the radius as ρ. The math is identical; only the names differ, so always confirm which convention a problem uses before plugging in.
How it’s calculated
Physics/ISO 80000-2 convention. Rectangular to spherical: r = √(x²+y²+z²), θ = acos(z/r) in [0°, 180°], φ = atan2(y, x) in (-180°, 180°]. Spherical to rectangular: x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ. Angles are entered and shown in degrees, with radians alongside.
At the origin (r = 0) the angles are undefined and reported as 0. The math convention that swaps θ and φ (and uses ρ for radius) gives the same point under different labels.
Rectangular and spherical (physics convention)
| (x, y, z) | r | θ (polar) | φ (azimuth) |
|---|---|---|---|
| (1, 1, 1) | 1.732 | 54.74° | 45° |
| (0, 0, 5) | 5 | 0° | 0° |
| (1, 0, 0) | 1 | 90° | 0° |
| (0, 1, 0) | 1 | 90° | 90° |
| (0, 0, -3) | 3 | 180° | 0° |
Computed with r = √(x²+y²+z²), θ = acos(z/r), φ = atan2(y, x).
Common mistakes
- Swapping θ and φ because a textbook uses the opposite convention.
- Using atan(y/x) instead of atan2(y, x), which loses the correct quadrant for φ.
- Measuring the polar angle θ up from the xy-plane instead of down from the +z axis.
- Mixing degrees and radians between the angle inputs and a trig function.
Frequently asked questions
How do you convert rectangular to spherical coordinates?
Use r = √(x²+y²+z²), θ = acos(z/r) for the polar angle, and φ = atan2(y, x) for the azimuth. This tool reports θ and φ in both degrees and radians.
What is the difference between the physics and math conventions?
Physics (ISO) uses θ for the polar angle and φ for the azimuth. Many calculus books swap them and write the radius as ρ. The point is the same; only the labels change. This tool uses the physics convention.
What are the formulas from spherical back to rectangular?
x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ, with θ the polar angle from +z and φ the azimuth from +x.
Why are the angles undefined at the origin?
When r = 0 the point sits at the origin, so no direction is singled out. Both θ and φ can be anything, and the tool reports 0.