Cylindrical Coordinates Calculator

Convert a Cartesian point (x, y, z) into cylindrical coordinates (r, θ, z) using r = √(x² + y²) and θ = atan2(y, x).

Quick Facts

Radius formula
r = √(x² + y²)
The straight-line distance from the z-axis to the point's projection on the xy-plane.
Angle formula
θ = atan2(y, x)
Uses the two-argument arctangent so θ lands in the correct quadrant; z carries over unchanged.

Your Results

Calculated
Radius (r)
-
Distance from the z-axis
Angle (θ)
-
Measured from the +x-axis
Height (z)
-
Unchanged from Cartesian z
Cylindrical point
-
(r, θ, z) notation

Ready

Enter x, y, and z, then press Calculate.

About the Cylindrical Coordinate System

Cylindrical coordinates describe a point in 3D space using three values: r, the radial distance from the z-axis to the point's projection on the xy-plane; θ (theta), the angle that projection makes with the positive x-axis; and z, the height above or below the xy-plane — identical to the Cartesian z. This calculator converts a Cartesian point (x, y, z) into its cylindrical equivalent (r, θ, z).

The conversion formulas

  • Radius: r = √(x² + y²) — the straight-line distance from the z-axis, found with the Pythagorean theorem.
  • Angle: θ = atan2(y, x) — the two-argument arctangent, which correctly places θ in the right quadrant for any sign combination of x and y.
  • Height: z = z — the vertical coordinate carries over unchanged.

The reverse conversion, from cylindrical back to Cartesian, uses x = r·cos(θ), y = r·sin(θ), and z = z.

Where cylindrical coordinates are used

Cylindrical coordinates are the natural choice whenever a system has rotational symmetry around an axis: pipes and cylinders in engineering, robotic-arm positioning, electromagnetic field problems, and cylindrical or polar plots in physics and CAD software. Because θ wraps around a circle, this system often simplifies equations that would be awkward in pure Cartesian (x, y, z) form.

Frequently Asked Questions

How do I convert Cartesian (x, y, z) to cylindrical (r, θ, z)?
Compute r = √(x² + y²) for the radius, θ = atan2(y, x) for the angle in the correct quadrant, and keep z unchanged. For example, (3, 4, 5) converts to r = 5, θ ≈ 53.13°, z = 5.
Why use atan2 instead of arctan(y/x)?
Plain arctan(y/x) can't distinguish between quadrants — it returns the same value for (3,4) and (-3,-4). atan2(y, x) uses the signs of both x and y to place θ in the correct quadrant, giving a result in the range (−180°, 180°].
What happens when x = 0 and y = 0?
The point lies exactly on the z-axis, so r = 0 and θ is undefined. This calculator reports θ as 0° by convention in that case; only z determines the point's position.
How is this different from spherical coordinates?
Cylindrical coordinates keep z as a linear height and only convert the xy-plane to polar form (r, θ). Spherical coordinates instead measure a single radial distance from the origin plus two angles. Cylindrical is preferred when a system has an axis of symmetry with a meaningful straight-line height, like a pipe or a cylinder.

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