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Cartesian to Spherical

Spherical coordinates (ρ,θ,φ): ρ = distance from origin; θ = azimuthal angle in xy-plane; φ = polar angle from z-axis. ρ = √(x²+y²+z²), θ = atan2(y,x), φ = arccos(z/ρ).

Concept Fundamentals
√(x²+y²+z²)
ρ
atan2(y,x)
θ
arccos(z/ρ)
φ
x=ρ sin φ cos θ, y=ρ sin φ sin θ, z=ρ cos φ
Inverse

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ρ ≥ 0; θ ∈ [0, 2π); φ ∈ [0, π]. Inverse: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ. Cylindrical r = ρ sin φ; cylindrical z = ρ cos φ.

Key quantities
√(x²+y²+z²)
ρ
Key relation
atan2(y,x)
θ
Key relation
arccos(z/ρ)
φ
Key relation
x=ρ sin φ cos θ, y=ρ sin φ sin θ, z=ρ cos φ
Inverse
Key relation

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Why: Spherical coordinates simplify problems with spherical symmetry: gravitational fields, atomic orbitals, antenna radiation patterns. Used in physics, astronomy, and 3D graphics.

How: ρ = √(x²+y²+z²), θ = atan2(y,x), φ = arccos(z/ρ). θ is azimuth; φ is polar angle from z-axis. At origin, ρ=0 and angles are undefined.

ρ ≥ 0; θ ∈ [0, 2π); φ ∈ [0, π].Inverse: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.
Sources:Khan Academy — Spherical CoordinatesWolfram MathWorld — Spherical Coordinates

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Convert CoordinatesEnter (x, y, z) to get (ρ, θ, φ)

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🧮 Fascinating Math Facts

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

— 3D Geometry

↩

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

— Conversion

Key Takeaways

  • • ρ = distance from origin: ρ=x2+y2+z2\rho = \sqrt{x^2 + y^2 + z^2}
  • • θ = azimuthal angle in xy-plane: θ=atan2(y,x)\theta = \text{atan2}(y, x)
  • • φ = polar angle from z-axis: ϕ=arccos⁡(z/ρ)\phi = \arccos(z/\rho)
  • • φ ∈ [0, π]: 0 at +z, π/2 in xy-plane, π at -z
  • • Ideal for spherical symmetry: gravity, orbitals, waves

Did You Know?

Physics & Gravity

Newton's law of gravitation and Coulomb's law are naturally expressed in spherical coordinates due to radial symmetry.

Quantum Mechanics

Hydrogen atom wavefunctions are written in spherical coordinates. The angular part uses spherical harmonics.

Astronomy

Celestial coordinates (right ascension, declination) are a spherical-like system for locating stars.

GPS & Earth

Latitude and longitude are essentially spherical coordinates on Earth's surface (with radius fixed).

Laplace Equation

Solutions to ∇²V=0 in spherical coordinates separate into radial and angular parts (Legendre polynomials).

Origin Special Case

At (0,0,0), ρ=0 but θ and φ are undefined. No unique direction from the origin.

Understanding Cartesian to Spherical

Spherical coordinates use: ρ (distance from origin), θ (angle in xy-plane from x-axis), φ (angle from +z-axis).

ρ=x2+y2+z2,θ=atan2(y,x),ϕ=arccos⁡(z/ρ)\rho = \sqrt{x^2 + y^2 + z^2}, \quad \theta = \text{atan2}(y,x), \quad \phi = \arccos(z/\rho)

φ = 0 points along +z; φ = π/2 lies in the xy-plane; φ = π points along -z.

Expert Tips

θ vs Cylindrical

θ in spherical is the same as θ in cylindrical — the azimuthal angle in the xy-plane.

φ Convention

Some texts swap θ and φ. This calculator uses the physics convention: φ from z-axis, θ in xy-plane.

Origin

At the origin, ρ=0. θ and φ are undefined. Many conventions use θ=φ=0 by default.

Inverse Formulas

x = ρ·sin(φ)·cos(θ), y = ρ·sin(φ)·sin(θ), z = ρ·cos(φ). Use Spherical to Cartesian calculator.

Frequently Asked Questions

What is ρ?

ρ (rho) is the radial distance from the origin to the point: ρ = √(x² + y² + z²).

What is the difference between θ and φ?

θ is the azimuthal angle in the xy-plane (like longitude). φ is the polar angle from the +z-axis (like colatitude).

What happens at the origin?

ρ = 0. θ and φ are undefined because there is no unique direction.

What is the range of φ?

φ ∈ [0, π]. φ = 0 at +z, φ = π/2 in the xy-plane, φ = π at -z.

How is this different from cylindrical?

Cylindrical uses r (distance from z-axis) and z. Spherical uses ρ (distance from origin) and φ (angle from z-axis).

When is spherical preferred?

Problems with spherical symmetry: point charges, gravitational fields, atomic orbitals, waves from a point source.

How do I convert back to Cartesian?

x = ρ·sin(φ)·cos(θ), y = ρ·sin(φ)·sin(θ), z = ρ·cos(φ). Use our Spherical to Cartesian calculator.

How to Use This Calculator

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

Note: Physics convention: φ measured from +z-axis. θ normalized to [0°, 360°). At origin, φ = 0 by convention.

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