Spherical to Cartesian Converter
Convert spherical coordinates to Cartesian (x, y, z) with standard formulas.
Convert coordinates
Result (Cartesian)
x: –
y: –
z: –
Tip: 3D graphics often use the math convention. Some physics and engineering texts invert θ and φ — select the right one above.
Need Cartesian → Spherical instead?Formulas used
There are multiple notation systems for spherical coordinates. This tool supports the two most common ones.
1. Math / Calculus convention (default)
Coordinates: (ρ, θ, φ)
- ρ = radial distance (ρ ≥ 0)
- θ = azimuth angle in x-y plane (from +x axis, toward +y)
- φ = polar/inclination angle from +z axis (0 to π)
x = ρ · sin(φ) · cos(θ)y = ρ · sin(φ) · sin(θ)z = ρ · cos(φ)
2. Physics / Engineering convention
Coordinates: (r, θ, φ)
- r = radial distance (r ≥ 0)
- θ = polar angle from +z axis
- φ = azimuth angle in x-y plane
x = r · sin(θ) · cos(φ)y = r · sin(θ) · sin(φ)z = r · cos(θ)
Degrees vs radians
Internally the calculator always converts your angles to radians using:
radians = degrees × π / 180
FAQ
Why are there two different angle orders?
Because different fields evolved differently. Many pure math / multivariable calculus books use (ρ, θ, φ) with θ in the plane and φ from z. Many physics and electromagnetism books swap the symbols. The vector you get is the same if you interpret the angles correctly.
What if I only know the polar angle from the z axis?
Select the convention that uses that angle as the “second” angle, and put the azimuth in the other field.
What units does the output use?
The converter returns x, y, z in the same linear unit as ρ (or r): if your radius was in meters, the result is in meters.
Can this be used for vector fields?
This page converts a single point. For full vector-field transformations you need the corresponding Jacobian/transformation rules — check your multivariable calculus or physics reference.
Formula (LaTeX) + variables + units
','\
x = ρ · sin(φ) · cos(θ) y = ρ · sin(φ) · sin(θ) z = ρ · cos(φ)
x = r · sin(θ) · cos(φ) y = r · sin(θ) · sin(φ) z = r · cos(θ)
radians = degrees × π / 180
- No variables provided in audit spec.
- Need Cartesian → Spherical instead? — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/cartesian-to-spherical
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.