GEOMETRYCoordinate GeometryMathematics Calculator
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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
Convert CoordinatesEnter Cartesian or cylindrical to convert

Why This Mathematical Concept Matters

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].
  • ●On z-axis: r=0, θ undefined.
Sources:Khan Academy — Cylindrical CoordinatesWolfram MathWorld — Cylindrical Coordinates

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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