How do you convert Cartesian to spherical coordinates?

For the math convention, first compute ρ = √(x² + y² + z²). Then θ = atan2(y, x) for the azimuth in the x-y plane, and φ = arccos(z / ρ) for the polar angle from the +z axis. If ρ = 0, the angles are undefined and can be set to 0.

What is the difference between math and physics spherical conventions?

In the math convention, θ is the azimuth in the x-y plane and φ is the polar angle from +z. In many physics texts, θ is the polar angle from +z and φ is the azimuth. This tool lets you pick which convention you want.

Can I get the angles in degrees?

Yes. Choose degrees in the options and the tool will convert the computed angles from radians to degrees.

Cartesian to Spherical Converter | Convert (x, y, z) to (ρ, θ, φ)

Convert coordinates

x Cartesian

y Cartesian

z Cartesian

Angle unit

Convention

Result (Spherical)

ρ / r: –

θ (azimuth): –

φ (polar from +z): –

Note: for math convention, θ is azimuth in x-y plane and φ is polar from +z; for physics convention, θ is polar from +z and φ is azimuth in x-y plane.

Need the inverse? Try Spherical to Cartesian.

Formulas used

First, compute the radial distance from the origin:

ρ = √(x² + y² + z²)

1. Math / Calculus convention

Coordinates: (ρ, θ, φ)

ρ: distance from origin
θ: azimuth in x-y plane = atan2(y, x)
φ: polar angle from +z = arccos(z / ρ)

ρ = √(x² + y² + z²)
θ = atan2(y, x)
φ = arccos(z / ρ)

2. Physics / Engineering convention

Coordinates: (r, θ, φ)

r: distance from origin (same as ρ)
θ: polar angle from +z = arccos(z / r)
φ: azimuth in x-y plane = atan2(y, x)

r = √(x² + y² + z²)
θ = arccos(z / r)
φ = atan2(y, x)

Degrees vs radians

The internal calculations use radians. If you select degrees, the tool converts using:

degrees = radians × 180 / π

FAQ

What if ρ = 0?

If x = y = z = 0, the point is the origin. Radius is 0 and angles are undefined. We return 0 for both angles for convenience.

Why two different names (ρ vs r, θ vs φ)?

Different textbooks and fields prefer different symbols. This tool lets you switch so you can match your source.

Will this work for negative x, y, or z?

Yes. Thanks to atan2, azimuth is always placed in the correct quadrant.