Convert between cylindrical (r, θ, z) and Cartesian (x, y, z) coordinates with step-by-step solutions, reference angle table, and top-down projection visual.
Cylindrical coordinates extend the familiar 2D polar system into three dimensions by adding a height axis z. A point in space is described by three values: the radial distance r from the z-axis, the angle θ measured counter-clockwise from the positive x-axis in the xy-plane, and the height z above (or below) the xy-plane. This system is especially convenient for problems with rotational symmetry around a central axis — think of pipes, solenoids, drill bits, or any geometry that naturally wraps around one direction.
Converting between cylindrical and Cartesian coordinates is straightforward. Going from cylindrical to Cartesian: x = r·cos θ, y = r·sin θ, and z stays the same. Going the other way: r = √(x² + y²), θ = atan2(y, x), and z is unchanged. The atan2 function is preferred over simple arctan because it correctly handles all four quadrants, returning an angle in the range [0, 2π).
This calculator performs both conversions instantly, shows each algebraic step, and renders a top-down projection of the point in the xy-plane. A reference table lists cosine and sine values for 17 standard angles so you can verify results by hand. Preset buttons let you explore special points such as the unit vectors along each axis and the classic 45° diagonal. Choose degrees or radians — the tool handles both seamlessly.
Converting between cylindrical (r, θ, z) and Cartesian (x, y, z) coordinates requires evaluating trig functions, choosing the right quadrant for atan2, and keeping track of whether angles are in degrees or radians. Errors in any one of these steps throw off the whole result. This calculator handles both conversion directions instantly, shows every algebraic step, and renders a top-down projection so you can confirm the geometry visually. Multivariable-calculus and physics students use it to check homework, and engineers verify coordinate transforms for cylindrical FEA meshes.
Cylindrical → Cartesian: x = r·cos(θ) y = r·sin(θ) z = z Cartesian → Cylindrical: r = √(x² + y²) θ = atan2(y, x) z = z Distance to origin = √(x² + y² + z²)
Result: x = 2.5, y ≈ 4.3301, z = 2
x = 5·cos(60°) = 5·0.5 = 2.5. y = 5·sin(60°) = 5·(√3/2) ≈ 4.3301. z remains 2.
Cylindrical coordinates are the natural choice whenever a problem has rotational symmetry about a single axis. A pipe, wire, solenoid, drill hole, or tree trunk is most simply described with r (how far from the axis), θ (how far around), and z (how far along). In multivariable calculus, triple integrals over cylinders and cones simplify dramatically in cylindrical form because the Jacobian dV = r dr dθ dz factors cleanly. Laplace's equation, the heat equation, and the wave equation all separate neatly in cylindrical coordinates, producing Bessel functions as radial solutions.
Gauss's law and Ampère's law frequently exploit cylindrical symmetry. The magnetic field around an infinitely long straight wire depends only on r: B = μ₀I/(2πr). The electric field inside a coaxial cable also depends only on r, making the problem one-dimensional when expressed in cylindrical coordinates. Computing the flux through a cylindrical Gaussian surface or the circulation around a circular Ampèrian loop becomes a single-variable integral, vastly simpler than the equivalent Cartesian calculation.
Cylindrical coordinates can be viewed as 2-D polar coordinates (r, θ) extruded along the z-axis. At z = 0 the two systems are identical. Spherical coordinates (ρ, φ, θ) relate to cylindrical by ρ = √(r² + z²) and φ = arctan(r/z). Moving between all three systems is a common exercise in vector calculus and is essential in physics: gravitational problems favour spherical, magnetic-coil problems favour cylindrical, and Cartesian coordinates remain king for rectangular geometries and most numerical simulations.
A 3D coordinate system using radial distance r, azimuthal angle θ, and height z to locate points in space. Use this as a practical reminder before finalizing the result.
θ is the angle in the xy-plane, measured counter-clockwise from the positive x-axis, typically ranging from 0 to 2π (or 0° to 360°). Keep this note short and outcome-focused for reuse.
atan2(y, x) returns the correct angle in all four quadrants, whereas arctan(y/x) only covers (−π/2, π/2) and fails when x = 0. Apply this check where your workflow is most sensitive.
Whenever the geometry has rotational symmetry about an axis — cylinders, pipes, circular plates, electromagnetic fields around a wire. Use this checkpoint when values look unexpected.
Cylindrical coordinates are polar coordinates (r, θ) plus a z-axis. In 2D, when z = 0, they reduce to standard polar coordinates.
First convert to Cartesian, then to spherical, or use ρ = √(r² + z²), φ = arctan(r/z), θ stays the same. Validate assumptions before taking action on this output.