Cylindrical Coordinates Calculator — Cartesian Conversion
Convert any point between cylindrical (ρ, θ, z) and Cartesian (x, y, z) coordinate systems. Enter the coordinates in either system and get the full conversion with the angle in degrees and radians.
Conversion direction
°
x = 3.5355, y = 3.5355, z = 3
- 1
θ in radians
45° × π ÷ 180 = 0.785398 - 2
y = ρ × sin θ
5 × sin(0.7854 rad) = 3.5355 - 3
x = ρ × cos θ
5 × cos(0.7854 rad) = 3.5355
How does this calculator work?
Cylindrical coordinates (ρ, θ, z) relate to Cartesian by x = ρ cos θ, y = ρ sin θ, z = z. Inverse: ρ = √(x²+y²), θ = atan2(y, x). Choose the direction, enter the coordinates, and get the result in degrees and radians instantly.
Formula
How this is calculated
Cylindrical coordinates describe a point in 3-D space using three values: the radial distance ρ (rho) from the z-axis (always ≥ 0), the azimuthal angle θ (theta) measured counterclockwise from the positive x-axis in the xy-plane, and the axial height z, which is identical to the Cartesian z-coordinate.
Converting from cylindrical to Cartesian uses the trigonometric identities x = ρ cos θ and y = ρ sin θ, with z unchanged. Converting back uses the inverse: ρ = √(x² + y²) (the Euclidean distance from the z-axis) and θ = atan2(y, x), which is the four-quadrant arctangent that correctly places the angle in the right quadrant regardless of the signs of x and y.
Cylindrical coordinates are natural for problems with rotational symmetry about an axis — electric fields around a wire, fluid flow in a pipe, or any object that looks the same when rotated. In such cases the math often simplifies compared with Cartesian form. This calculator works with θ in degrees for input convenience and also reports the radian value.
Frequently asked questions
Cylindrical coordinates (ρ, θ, z) use a radial distance from the z-axis and an axial height z. Spherical coordinates (r, θ, φ) use a radial distance from the origin and two angles. Cylindrical has one linear axis (z) remaining; spherical does not.
The simple arctan function cannot distinguish angles in opposite quadrants because arctan(y/x) gives the same value for (y, x) and (−y, −x). The atan2(y, x) function takes both signs into account and returns angles in the full −180° to +180° range, giving the correct quadrant every time.
First convert to Cartesian: x = ρ cos θ, y = ρ sin θ, z = z. Then convert to spherical: r_sph = √(ρ² + z²), polar angle φ = atan2(ρ, z), azimuthal angle = same θ.
TG we-Calculate Editorial Team. (2026). Cylindrical Coordinates Calculator — Cartesian Conversion [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cylindrical-coordinates-calculator
TG we-Calculate Editorial Team. "Cylindrical Coordinates Calculator — Cartesian Conversion." TG we-Calculate. 2026. https://we-calculate.com/calculator/cylindrical-coordinates-calculator.
TG we-Calculate Editorial Team, "Cylindrical Coordinates Calculator — Cartesian Conversion," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cylindrical-coordinates-calculator
@misc{wecalculate_cylindrical_coordinates_calculator, title = {Cylindrical Coordinates Calculator — Cartesian Conversion}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cylindrical-coordinates-calculator}}, year = {2026}, note = {TG we-Calculate} }
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