Convert between cylindrical and Cartesian coordinates in 3D space
Last updated: April 2026 | By Patchworkr Team
ρ = √(x² + y²)
φ = arctan(y/x)
z = z
x = ρ cos(φ)
y = ρ sin(φ)
z = z
ρ ≥ 0 (radial distance, always non-negative)
0 ≤ φ < 360° or 0 ≤ φ < 2π rad (azimuthal angle)
-∞ < z < ∞ (height, unbounded)
Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates into 3D space. A point in space is described by three values: ρ (rho, the radial distance from the z-axis), φ (phi, the azimuthal angle from the positive x-axis in the xy-plane), and z (the height above or below the xy-plane). This system is particularly useful for problems with cylindrical symmetry.
The radial distance ρ tells you how far the point is from the z-axis (the vertical axis), similar to how radius works in polar coordinates. The angle φ tells you the direction in the horizontal plane, measured counterclockwise from the positive x-axis. The z-coordinate is identical to the z-coordinate in Cartesian coordinates, representing vertical position.
Cylindrical coordinates are extensively used in physics, engineering, and mathematics when dealing with problems involving cylinders, circular motion, or axial symmetry. Examples include electromagnetic field calculations around wires, fluid flow in pipes, heat transfer in cylindrical objects, robotics (for cylindrical work envelopes), and CAD/CAM systems for machining cylindrical parts.
Step 1: Choose your conversion direction. Select "Cartesian → Cylindrical" if you have (x, y, z) coordinates and want to convert to (ρ, φ, z), or select "Cylindrical → Cartesian" for the reverse.
Step 2 (Cartesian to Cylindrical): Enter the x, y, and z coordinates. The calculator will compute ρ (the distance from the z-axis), φ (the angle in the xy-plane), and z (which remains unchanged).
Step 2 (Cylindrical to Cartesian): Enter ρ (radial distance, must be non-negative), φ (angle in degrees, typically 0-360), and z. The calculator will compute the corresponding x, y, and z Cartesian coordinates.
Technical Details: When converting from Cartesian to cylindrical, the calculator uses atan2(y, x) to properly handle all quadrants and provides φ in the range [0, 360°). When converting from cylindrical to Cartesian, the angle φ is converted from degrees to radians before applying the cosine and sine functions. The z-coordinate is preserved in both directions, making cylindrical coordinates particularly convenient for problems with vertical symmetry.
Scenario: A robotics engineer is programming a cylindrical robot arm. A sensor detects an object at Cartesian position (5, 5, 10) cm. The robot needs cylindrical coordinates to move to this position.
Converted to Cylindrical:
ρ (radial): 7.071 cm (extend arm this distance from center)
φ (angle): 45° (rotate base to this angle)
z (height): 10 cm (raise arm to this height)
The robot's control system naturally works in cylindrical coordinates because it has a rotating base (φ), an extending arm (ρ), and a vertical lift (z). Converting from Cartesian sensor data to cylindrical robot commands enables precise positioning in the robot's native coordinate system.
Cylindrical uses (ρ, φ, z): distance from z-axis, angle, and height. Spherical uses (r, θ, φ): distance from origin, polar angle, and azimuthal angle. Cylindrical is best for cylindrical symmetry, spherical for spherical symmetry.
ρ represents distance from the z-axis, which is always positive or zero. Negative distances don't make physical sense. Different angles (φ) handle points in all directions around the axis.
Use cylindrical coordinates when your problem has rotational symmetry around a vertical axis: pipes, cylinders, circular motion, axial symmetry, rotating machinery, or problems naturally described by radius and angle.
The calculator uses atan2(y, x) which correctly handles all four quadrants. Points with negative x or y will have φ values between 90° and 270°, automatically placing them in the correct angular position.
Yes, z can be positive, negative, or zero. Positive z is above the xy-plane, negative z is below it, and z=0 is in the plane. The z-coordinate works exactly like in Cartesian coordinates.
Angles are periodic: 370° = 10°, 720° = 0°, etc. The calculator normalizes φ to [0, 360°) for consistency. You can enter any angle; the result will be equivalent but in the standard range.
Cylindrical coordinates simplify equations for electromagnetic fields around wires, fluid flow in pipes, heat distribution in cylinders, and angular momentum problems. Many differential equations are easier in cylindrical form for axially symmetric systems.
Cylindrical coordinates are polar coordinates (ρ, φ) in the xy-plane plus a z-component. If z = 0, cylindrical coordinates reduce exactly to 2D polar coordinates.
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