Free Cylindrical Coordinates Calculator

Last updated October 12, 2026 · Report an error

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Cylindrical coordinates (r, θ, z) describe a point in 3D space using a radius r and angle θ in the xy-plane — exactly like 2D polar coordinates — plus a straight-line height z above or below that plane. They're the natural middle ground between Cartesian (x, y, z) coordinates, which treat all three axes identically, and spherical coordinates, which replace both the radius and height with two angles and a single distance from the origin. Cylindrical coordinates are the tool of choice whenever a problem has rotational symmetry around a single axis: physics and engineering use them to describe the electric or magnetic field around a wire, the flow of fluid through a pipe, or the temperature distribution in a cylindrical rod; robotics and CNC machining use them to describe the reach of an arm that rotates around a vertical axis and extends outward and upward; and multivariable calculus uses them to simplify triple integrals over cylindrical or conical regions that would be far messier in plain Cartesian form.

How it works

Choose a conversion direction, then enter either (x, y, z) or (r, θ, z). Converting from Cartesian to cylindrical treats x and y exactly like a 2D polar-coordinate conversion: r = √(x² + y²) gives the straight-line distance from the z-axis, and θ = atan2(y, x) gives the angle around that axis, measured from the positive x-axis. The height z is left completely unchanged in both directions, since cylindrical coordinates only reinterpret the flat xy-plane, not the vertical axis. Converting the other way, from cylindrical back to Cartesian, uses x = r·cos(θ) and y = r·sin(θ), again with z passed through as-is.

Enter conversion direction.
Enter x (for Cartesian → Cylindrical).
Enter y (for Cartesian → Cylindrical).
Enter z (for Cartesian → Cylindrical).
Enter r (for Cylindrical → Cartesian).
Enter θ in degrees (for Cylindrical → Cartesian).
Enter z (for Cylindrical → Cartesian).
Click Calculate to see your results.

Examples

(3, 4, 5) to cylindrical

r = √(3²+4²) = √25 = 5, θ = atan2(4, 3) ≈ 53.13°, and z = 5 unchanged — so the cylindrical coordinates are (5, 53.13°, 5).

(r=10, θ=30°, z=-2) to Cartesian

x = 10·cos(30°) = 10×0.866 ≈ 8.66, y = 10·sin(30°) = 10×0.5 = 5, and z = -2 unchanged — so the Cartesian coordinates are (8.66, 5, -2).

(0, 0, 7) to cylindrical

r = √(0²+0²) = 0 and θ is undefined (any angle works when r = 0, since the point sits exactly on the z-axis), while z = 7 unchanged — a reminder that angle becomes meaningless right on the central axis.

Common mistakes to avoid

Forgetting that z passes through unchanged in both conversion directions — it's not affected by the r/θ calculation at all.
Using a plain arctan(y/x) instead of atan2(y, x), which can silently place θ in the wrong quadrant.
Reporting a specific θ value for a point that lies exactly on the z-axis, where angle is genuinely undefined.
Mixing up degrees and radians for θ, especially when feeding the result into another formula or piece of software that expects one specific unit.

Frequently asked questions

Cylindrical coordinates keep z as a straight vertical height and use a radius measured only within the flat xy-plane. Spherical coordinates instead use a single radius measured all the way from the origin to the point, plus two angles — one around the z-axis and one down from it.

A plain arctan(y/x) can't distinguish between a point in the first quadrant and the diametrically opposite point in the third quadrant, since the ratio y/x is identical for both. atan2(y, x) is given x and y as separate values, so it can correctly place θ anywhere in the full 360° range.

When x = 0 and y = 0, r is also 0, and θ becomes undefined — geometrically, every angle describes the same point (the axis itself), so no single angle is more correct than any other.

By convention, no — r is defined as √(x²+y²), which is always non-negative. Some specialized contexts allow a signed r paired with an adjusted angle, but this calculator follows the standard non-negative convention.

No — z is identical in both coordinate systems and simply passes through unchanged in either direction of the conversion, since cylindrical coordinates only ever modify how x and y (not z) are represented.

They're standard in describing anything with an axis of rotation: pipe flow and heat conduction in engineering, electromagnetic fields around wires and solenoids in physics, and the working envelope of a robotic arm or lathe that rotates around a fixed vertical axis.