Spherical coordinates suit anything with a centre: planets, antennas, atomic orbitals, sound fields and geographic positions (latitude is 90° − θ, longitude is φ).
How to use the Spherical Coordinates Calculator
- Conversion
- x
- y
- z
- r
- θ (polar angle from z)
- φ (azimuth)
Results update as you type, and everything is calculated in your browser.
How it works
This calculator uses the physics convention (θ from the z-axis, φ in the xy-plane). Swap the names if your textbook uses the mathematics convention.
Inputs explained
| Field | What to enter |
|---|---|
| Conversion | Choose one: Cartesian → spherical, Spherical → Cartesian |
| x | A number |
| y | A number |
| z | A number |
| r | A number |
| θ (polar angle from z) | Units: °, rad, π rad, grad, turn |
| φ (azimuth) | Units: °, rad, π rad, grad, turn |
What each result means
- r = sqrt([x]^2 + [y]^2 + [z]^2)
- θ (polar angle) = deg(acos([z]÷sqrt([x]^2 + [y]^2 + [z]^2)))
- φ (azimuth) = deg(atan2([y], [x]))
- x = [r]×sin(rad([θ (polar angle from z)]))×cos(rad([φ (azimuth)]))
- y = [r]×sin(rad([θ (polar angle from z)]))×sin(rad([φ (azimuth)]))
- z = [r]×cos(rad([θ (polar angle from z)]))
Example values
How the main result changes when x changes and everything else stays at the example values above:
| x | r |
|---|---|
| 0.5 | 2.8723 |
| 0.75 | 2.9262 |
| 1 | 3 |
| 1.25 | 3.0923 |
| 1.5 | 3.2016 |
| 2 | 3.4641 |
Formula
r = √(x² + y² + z²), θ = arccos(z/r), φ = atan2(y, x)
x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ
Worked example
Inputs — Conversion: Cartesian → spherical; x: 1; y: 2; z: 2.
Results — r = 3; θ (polar angle) = 48.189685 °; φ (azimuth) = 63.434949 °.
Key terms
- Azimuth
- The angle in the xy-plane from the x-axis.
Frequently asked questions
Which angle is which?
Physics: θ from the z-axis, φ around it. Maths textbooks often swap them.
How do spherical coordinates relate to latitude and longitude?
Latitude = 90° − θ, longitude = φ.
What is the volume element?
dV = r² sin θ dr dθ dφ.
What is r for the origin?
0, and the angles are undefined.