Cylindrical to Cartesian Converter
Convert cylindrical coordinates (r, θ, z) to Cartesian coordinates (x, y, z). Supports degrees and radians, shows formulas, and is suitable for multivariable calculus, physics, and engineering.
Cylindrical inputs
Distance from the z-axis; must be non-negative.
Measured in the x-y plane from the +x axis, counterclockwise.
Same value as the cylindrical z coordinate.
z is copied directly from the cylindrical coordinate while x and y rotate by θ. Negative r values are not permitted.
How to Use This Converter
Enter the radial distance, throw in the angle, and keep the height as-is. Switch the angle unit between degrees and radians depending on your source material, then hit Convert or wait for the inputs to settle. The Cartesian result updates without refreshing the page.
Methodology
This tool first converts θ into radians (if the input was degrees) and then applies the standard polar-plane transformations: x = r × cos(θ) and y = r × sin(θ). z is carried forward unchanged because cylindrical coordinates already align with the Cartesian z-axis.
- The radial distance must be zero or positive; the calculator surfaces an error otherwise.
- Degrees are multiplied by π/180 before cosine and sine are evaluated.
- Results render with six decimal places to match the legacy formatting from the original converter.
Full original guide (expanded)
Definition
Cylindrical coordinates extend polar coordinates with a vertical height. The triplet (r, θ, z) describes a point by radial distance from the z-axis, the azimuthal angle in the x-y plane, and the elevation.
- r: radial distance from the z-axis (r ≥ 0)
- θ: angle measured from the +x axis, sweeping counterclockwise through the x-y plane
- z: same height value as in Cartesian coordinates
Conversion to Cartesian
x = r · cos(θ)y = r · sin(θ)z = z
Degrees to radians
If your angle is given in degrees, convert it first to keep cosine and sine accurate:
θrad = θdeg × π / 180
Relationship to other systems
Cylindrical coordinates are (r, θ, z). Drop z and you have polar coordinates (r, θ) in the plane. Switch to spherical coordinates (ρ, θ, φ) when you need the distance from the origin and an additional polar angle from the z-axis.
FAQ
Why do we use cylindrical coordinates?
They simplify problems with rotational symmetry around the z-axis like fluid flow in pipes, electromagnetic fields around wires, or cylindrical objects.
What happens if r is 0?
The point sits on the z-axis, so x = 0 and y = 0 regardless of θ. z stays as entered and controls the height above or below the plane.
My angle is 450° — is that ok?
Yes. Cosine and sine reduce any real angle to the equivalent position in the unit circle, so 450° works just like 90°.