How do you convert spherical to Cartesian coordinates?

For the common math convention (ρ, θ, φ) where θ is the azimuth in the x-y plane and φ is the polar angle from the +z axis, the formulas are: x = ρ · sin(φ) · cos(θ), y = ρ · sin(φ) · sin(θ), z = ρ · cos(φ).

What is the difference between math and physics conventions?

In math, φ is usually the angle from the +z axis (polar) and θ is the azimuth. In many physics texts the symbols are swapped: θ is the polar angle from +z, and φ is the azimuth. The converter lets you pick the convention.

Can I enter angles in degrees?

Yes, this tool supports both degrees and radians. Select the unit from the dropdown and the converter will interpret your angles correctly.

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Spherical to Cartesian Converter

Convert spherical coordinates (ρ, θ, φ) to Cartesian (x, y, z) with support for different angle conventions (math vs physics) and degrees/radians. Ideal for 3D math, engineering, physics and computer graphics.

Convert coordinates

ρ (radius) distance

θ / φ (first angle) (azimuth)

φ / θ (second angle) (polar from +z)

Angle unit

Convention

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.