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Cartesian to Cylindrical

Cylindrical coordinates (r,θ,z) extend 2D polar to 3D. r = √(x²+y²) is distance from z-axis; θ = atan2(y,x) is azimuthal angle; z stays unchanged. Ideal for pipes, wires, and rotational symmetry.

Concept Fundamentals
√(x²+y²)
r
atan2(y,x)
θ
z unchanged
z
x=r cos θ, y=r sin θ
Inverse

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Same r and θ formulas as 2D polar in the xy-plane. D in plane equation Ax+By+Cz+D=0 does not affect angle. Inverse: x = r·cos(θ), y = r·sin(θ), z = z.

Key quantities
√(x²+y²)
r
Key relation
atan2(y,x)
θ
Key relation
z unchanged
z
Key relation
x=r cos θ, y=r sin θ
Inverse
Key relation

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Why: Cylindrical coordinates simplify problems with axial symmetry: electric fields around wires, pipe flow, pressure vessels. The xy-plane is treated as 2D polar; z is independent.

How: r = √(x²+y²), θ = atan2(y,x), z = z. Use atan2 for correct quadrant. On the z-axis (x=y=0), r=0 and θ is undefined.

Same r and θ formulas as 2D polar in the xy-plane.D in plane equation Ax+By+Cz+D=0 does not affect angle.
Sources:Khan Academy — Cylindrical CoordinatesWolfram MathWorld — Cylindrical Coordinates

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Convert CoordinatesEnter (x, y, z) to get (r, θ, z)

Enter Cartesian Coordinates

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🧮 Fascinating Math Facts

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r = √(x²+y²), θ = atan2(y,x), z = z.

— 3D Geometry

↩

Inverse: x = r cos θ, y = r sin θ.

— Conversion

Key Takeaways

  • • Cylindrical = polar + z: r=x2+y2r = \sqrt{x^2 + y^2}, θ=atan2(y,x)\theta = \text{atan2}(y,x), z=zz = z
  • • r is the radial distance from the z-axis in the xy-plane
  • • θ is the azimuthal angle in the xy-plane from the x-axis
  • • z is unchanged — identical to Cartesian z
  • • Ideal for problems with cylindrical symmetry: pipes, wires, rotating fluids

Did You Know?

Physics & E&M

Electric fields around infinite wires and magnetic fields in solenoids are naturally expressed in cylindrical coordinates.

Fluid Dynamics

Pipe flow and axial flow problems use cylindrical coordinates. The Navier-Stokes equations simplify in cylindrical form.

Engineering

Pressure vessels, shafts, and cylindrical structures are analyzed using cylindrical coordinates for stress and strain.

Extension of Polar

Cylindrical coordinates are simply polar (r,θ) in the xy-plane plus z. The xy-plane is treated as 2D polar.

Laplace Equation

The Laplace equation in cylindrical coordinates separates variables, enabling solutions for potential fields.

z-Axis Special Case

When x = y = 0, r = 0 and θ is undefined. The point lies on the z-axis.

Understanding Cartesian to Cylindrical

Cylindrical coordinates extend 2D polar to 3D. The (x,y) plane is converted to (r,θ) with the same formulas as polar; z stays the same.

r=x2+y2,θ=atan2(y,x),z=zr = \sqrt{x^2 + y^2}, \quad \theta = \text{atan2}(y, x), \quad z = z

Use atan2(y,x) to get the correct quadrant for θ.

Expert Tips

Same as Polar

The r and θ formulas are identical to 2D polar. Only the xy-plane is considered; z is independent.

z-Axis

On the z-axis (x=y=0), r=0 but θ is undefined. Many conventions use θ=0 by default.

Angle Range

θ is typically normalized to [0, 2π) or (-π, π]. This calculator uses [0°, 360°).

Inverse Conversion

To go back: x = r·cos(θ), y = r·sin(θ), z = z. Use the Cylindrical to Cartesian calculator.

Frequently Asked Questions

What is the formula for r?

r = √(x² + y²). It is the perpendicular distance from the z-axis to the point.

Why does z stay the same?

z measures height above the xy-plane in both Cartesian and cylindrical. The coordinate systems share the same z-axis.

What happens on the z-axis?

When x = y = 0, r = 0. The angle θ is undefined because there is no unique direction in the xy-plane.

How is this different from spherical?

Cylindrical uses (r,θ,z): r is distance from z-axis. Spherical uses (ρ,θ,φ): ρ is distance from origin.

When is cylindrical preferred?

Problems with cylindrical symmetry: wires, pipes, rotating cylinders, axial flow.

Can I convert cylindrical to Cartesian?

Yes: x = r·cos(θ), y = r·sin(θ), z = z. Use our Cylindrical to Cartesian calculator.

What is the unit vector for r?

In cylindrical coordinates, the radial unit vector is (cos θ, sin θ, 0) in Cartesian components.

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 cylindrical (r, θ, z).
  3. Review the step-by-step solution and metrics.
  4. Use Copy Results to share.

Note: Angles are normalized to [0°, 360°). Standard floating-point arithmetic applies.

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