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Spherical to Cartesian

Convert (ρ, θ, φ) to (x, y, z): x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ. φ is from +z-axis; θ is in xy-plane. Used in physics, 3D graphics, and orbital mechanics.

Concept Fundamentals
x = ρ sin φ cos θ
x
y = ρ sin φ sin θ
y
z = ρ cos φ
z
ρ=1 → unit sphere
Unit

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ρ·sin φ projects onto xy-plane. Unit sphere: (sin φ cos θ, sin φ sin θ, cos φ). φ=0 → +z; φ=π → −z.

Key quantities
x = ρ sin φ cos θ
x
Key relation
y = ρ sin φ sin θ
y
Key relation
z = ρ cos φ
z
Key relation
ρ=1 → unit sphere
Unit
Key relation

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Why: Spherical-to-Cartesian conversion is essential for wave functions, field vectors, 3D camera positioning, and converting orbital elements to Cartesian positions.

How: Apply x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ. Angles must be in radians for cos/sin. When ρ=0, (x,y,z)=(0,0,0). ρ·sin φ is distance from z-axis.

ρ·sin φ projects onto xy-plane.Unit sphere: (sin φ cos θ, sin φ sin θ, cos φ).
Sources:Khan Academy — Spherical CoordinatesWolfram MathWorld — Spherical Coordinates

Run the calculator when you are ready.

Convert Spherical to CartesianEnter ρ, θ, φ

Enter Spherical Coordinates

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.

— Conversion

○

ρ=1 parametrizes unit sphere.

— Geometry

Key Takeaways

  • • x = ρ · sin(φ) · cos(θ), y = ρ · sin(φ) · sin(θ), z = ρ · cos(φ)
  • • θ and φ must be in radians when evaluating trig functions
  • • φ = 0 → point on +z-axis; φ = π/2 → point in xy-plane; φ = π → point on -z-axis
  • • When ρ = 0, (x, y, z) = (0, 0, 0) regardless of angles
  • • The factor sin(φ) projects the radial distance onto the xy-plane

Did You Know?

Geometry

ρ·sin(φ) is the perpendicular distance from the z-axis — the "r" in cylindrical coordinates.

Unit Sphere

When ρ = 1, (sin φ cos θ, sin φ sin θ, cos φ) parametrizes the unit sphere.

Physics

Spherical-to-Cartesian is used to convert wave functions, field vectors, and orbital positions.

3D Graphics

Spherical coordinates are used for camera positioning, environment mapping, and skyboxes.

Inverse Formulas

ρ = √(x²+y²+z²), θ = atan2(y,x), φ = arccos(z/ρ). Use Cartesian to Spherical calculator.

Jacobian

The volume element in spherical coordinates is ρ² sin(φ) dρ dθ dφ. The sin(φ) appears in the conversion.

Understanding Spherical to Cartesian

Project the point: first project onto the z-axis (z = ρ·cos φ), then the xy-component has length ρ·sin φ. Resolve that into x and y using θ.

x=rhosinphicostheta,y=rhosinphisintheta,z=rhocosphix = \\rho \\sin\\phi \\cos\\theta, \quad y = \\rho \\sin\\phi \\sin\\theta, \quad z = \\rho \\cos\\phi

Ensure θ and φ are in radians when evaluating cos and sin.

Expert Tips

Radians Required

cos and sin expect radians. Convert degrees: angle_rad = angle_deg × π/180.

φ = 0 and φ = π

When φ = 0, sin φ = 0 so x = y = 0; point on +z. When φ = π, point on -z.

Verify with ρ

Check: x² + y² + z² = ρ². Use this to verify your conversion.

θ vs φ

θ is in the xy-plane (azimuth). φ is from the z-axis (polar/colatitude). Don't confuse them.

Frequently Asked Questions

What are the conversion formulas?

x = ρ·sin(φ)·cos(θ), y = ρ·sin(φ)·sin(θ), z = ρ·cos(φ). Angles must be in radians.

Can I use degrees?

Yes. Convert first: θ_rad = θ_deg × π/180, φ_rad = φ_deg × π/180. This calculator supports both modes.

What happens when ρ = 0?

You get (0, 0, 0) — the origin. The angles do not affect the result.

What is φ = 90°?

φ = π/2 means the point lies in the xy-plane. z = ρ·cos(π/2) = 0.

How do I convert Cartesian to spherical?

ρ = √(x²+y²+z²), θ = atan2(y,x), φ = arccos(z/ρ). See our Cartesian to Spherical calculator.

Why sin(φ) for x and y?

ρ·sin(φ) is the distance from the z-axis. That distance is then resolved into x and y using cos(θ) and sin(θ).

What convention does this use?

Physics convention: φ measured from +z-axis (0 at north pole, π at south pole). θ in xy-plane from x-axis.

How to Use This Calculator

  1. Enter spherical (ρ, θ, φ). Select degrees or radians for the angles.
  2. Click a sample example to auto-fill and calculate.
  3. Click Calculate to get Cartesian (x, y, z).
  4. Review the step-by-step solution and use Copy Results to share.

Note: Physics convention: φ from +z-axis. ρ must be non-negative. Standard floating-point arithmetic.

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