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Cylindrical Coordinates

Cylindrical (r,θ,z): r = distance from z-axis, θ = azimuth in xy-plane, z = height. Cartesian↔Cylindrical: x=r cos θ, y=r sin θ, z=z; r=√(x²+y²), θ=atan2(y,x).

Concept Fundamentals
x=r cos θ, y=r sin θ, z=z
To Cartesian
r=√(x²+y²), θ=atan2(y,x), z=z
To Cylindrical
Distance from z-axis
r
Azimuthal angle
θ

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Same as 2D polar in xy-plane plus z. r ≥ 0; θ typically in [0, 2π) or (-π, π]. On z-axis: r=0, θ undefined.

Key quantities
x=r cos θ, y=r sin θ, z=z
To Cartesian
Key relation
r=√(x²+y²), θ=atan2(y,x), z=z
To Cylindrical
Key relation
Distance from z-axis
r
Key relation
Azimuthal angle
θ
Key relation

Ready to run the numbers?

Why: Cylindrical coordinates simplify problems with axial symmetry: electric fields around wires, pipe flow, rotating systems. Bidirectional conversion is common in physics and engineering.

How: Cartesian→Cylindrical: r=√(x²+y²), θ=atan2(y,x), z=z. Cylindrical→Cartesian: x=r cos θ, y=r sin θ, z=z. Use radians or degrees consistently.

Same as 2D polar in xy-plane plus z.r ≥ 0; θ typically in [0, 2π) or (-π, π].
Sources:Khan Academy — Cylindrical CoordinatesWolfram MathWorld — Cylindrical Coordinates

Run the calculator when you are ready.

Convert CoordinatesEnter Cartesian or cylindrical to convert

Sample Examples

Input

Cartesian (x, y, z)

Cylindrical (r, θ, z)

Results

Cartesian

(3, 4, 5)

Cylindrical

r = 5, θ = 53.13°, z = 5

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

🧮 Fascinating Math Facts

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x=r cos θ, y=r sin θ, z=z; r=√(x²+y²), θ=atan2(y,x).

— 3D Geometry

z

z unchanged in both systems.

— Property

Key Takeaways

  • r = √(x²+y²), θ = atan2(y,x), z = z
  • x = r cos θ, y = r sin θ, z = z
  • Polar + height: extends 2D polar to 3D
  • r ≥ 0; θ in [0, 2π)
  • Used for cylindrical symmetry

Did You Know?

  • Pipes, cylinders, rotational symmetry
  • Volume element: r dr dθ dz
  • Laplace in cylindrical coordinates
  • Bessel functions arise naturally
  • Used in E&M, fluid dynamics
  • Between polar and spherical

Understanding

Cylindrical = polar (r,θ) in xy-plane + z height.

r=x2+y2,θ=arctan⁡2(y,x),z=zr = \sqrt{x^2+y^2}, \quad \theta = \arctan2(y,x), \quad z = z
x=rcos⁡θ,y=rsin⁡θ,z=zx = r\cos\theta, \quad y = r\sin\theta, \quad z = z

Expert Tips

  • On z-axis: r=0, θ undefined
  • Same as polar when z=0
  • r is distance from z-axis
  • θ measured from +x axis

FAQ

Q: Relation to polar?
A: Polar + z; same (r,θ) in xy-plane.
Q: Applications?
A: Pipes, antennas, rotational symmetry.
Q: Volume element?
A: dV = r dr dθ dz.
Q: On z-axis?
A: r=0, θ arbitrary.
Q: Relation to spherical?
A: r = ρ sin φ, z = ρ cos φ.
Q: Gradient form?
A: ∇f = (∂f/∂r)r̂ + (1/r)(∂f/∂θ)θ̂ + (∂f/∂z)ẑ.
Q: When to use?
A: Cylindrical symmetry problems.

How to Use

  1. Enter Cartesian (x,y,z) or Cylindrical (r,θ,z)
  2. Other form updates automatically
  3. θ in degrees or radians

Disclaimer

r ≥ 0.

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