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Cylindrical Coordinates Calculator

Free cylindrical coordinates calculator. Convert Cartesian (x, y, z) to cylindrical (r, θ, z) or back, with radians and degrees supported.

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What Are Cylindrical Coordinates?

Cylindrical coordinates describe any point in three-dimensional space using three values: a radial distance r, an azimuthal angle θ, and a vertical height z. This system extends polar coordinates into 3D by keeping the z-axis unchanged, making it the natural framework for problems with rotational symmetry around a central axis — such as pipes, coils, pressure vessels, and electromagnetic waveguides.

Conversion Formulas

Cartesian (x, y, z) to Cylindrical (r, θ, z)

To convert a point from Cartesian to cylindrical coordinates, apply the following three relations:

  • r = √(x² + y²) — the perpendicular distance from the z-axis to the point; always non-negative.
  • θ = atan2(y, x) — the azimuthal angle measured counter-clockwise from the positive x-axis, returning a value in (−π, π] radians.
  • z = z — the vertical component is identical in both systems and requires no conversion.

Cylindrical (r, θ, z) to Cartesian (x, y, z)

To recover Cartesian coordinates from a cylindrical triple, apply the inverse relations:

  • x = r cos θ
  • y = r sin θ
  • z = z

Geometric Derivation

The derivation begins in the xy-plane. Any point (x, y) lies at a Euclidean distance r = √(x² + y²) from the origin, by the Pythagorean theorem. The angle θ satisfies cos θ = x/r and sin θ = y/r, immediately yielding x = r cos θ and y = r sin θ. Appending the unchanged z-coordinate elevates the 2D polar system into full cylindrical form. The two-argument atan2(y, x) function is used in place of a plain arctan to correctly resolve the quadrant for all (x, y) pairs — unlike arctan(y/x), which is undefined when x = 0 and returns ambiguous results when x < 0. According to Pauls Online Math Notes — Calculus III: Cylindrical Coordinates, this system is the standard choice whenever the geometry of a problem has an axis of symmetry.

Variable Reference

  • r (Radial Distance): The shortest distance from the z-axis to the point, computed as √(x² + y²). Always r ≥ 0. When r = 0, the point lies on the z-axis and θ is undefined.
  • θ (Azimuthal Angle): The angle swept counter-clockwise from the positive x-axis in the xy-plane. This calculator supports radians (range −π to π) or degrees (range −180° to 180°). Radians are the standard in calculus and physics; degrees are common in engineering and navigation.
  • z (Height): The signed vertical distance along the z-axis. This coordinate is identical in Cartesian and cylindrical systems — no formula is needed.

Worked Examples

Example 1: Cartesian to Cylindrical

Convert the Cartesian point (3, 4, 7) to cylindrical coordinates:

  • r = √(3² + 4²) = √(9 + 16) = √25 = 5
  • θ = atan2(4, 3) ≈ 0.9273 rad (≈ 53.13°)
  • z = 7

Result: (r, θ, z) = (5, 0.9273 rad, 7). Verification: x = 5 cos(0.9273) ≈ 3 ✓, y = 5 sin(0.9273) ≈ 4 ✓.

Example 2: Cylindrical to Cartesian

Convert the cylindrical point (6, π/3 rad, −2) to Cartesian:

  • x = 6 cos(π/3) = 6 × 0.5 = 3
  • y = 6 sin(π/3) = 6 × (√3/2) ≈ 5.196
  • z = −2

Real-World Applications

Cylindrical coordinates reduce complexity wherever circular or axial symmetry appears:

  • Integral Calculus: Triple integrals over cylinders and cones use the cylindrical volume element r dr dθ dz. The factor r is the Jacobian determinant of the transformation. As detailed in Whitman College — 15.2 Double Integrals in Cylindrical Coordinates, integrating over a cylinder of radius R and height H gives ∫₀ᴴ ∫₀²π ∫₀ᴿ r dr dθ dz = πR²H, recovering the standard volume formula.
  • Electromagnetism: The magnetic field around a long straight current-carrying wire is purely azimuthal: B = (μ₀I)/(2πr) in the θ̂-direction — a single-component description impossible without cylindrical coordinates.
  • Structural Engineering: Axisymmetric finite-element analyses of pressure vessels, nuclear fuel rods, and rotating shafts adopt RZ cylindrical formulations to collapse 3D problems into 2D, dramatically reducing computational cost.
  • Fluid Dynamics: Poiseuille flow in a circular pipe and Taylor-Couette flow between coaxial cylinders have velocity fields that depend only on r, reducing the Navier-Stokes equations to ordinary differential equations in a single variable.

Reference

Frequently asked questions

What is r in cylindrical coordinates?
In cylindrical coordinates, r is the radial distance from the z-axis to the point, computed as r = √(x² + y²) from the Cartesian components x and y. It represents the length of the perpendicular dropped from the point to the central axis. The value of r is always non-negative; a point located exactly on the z-axis has r = 0, and the angle θ is undefined in that case.
Why does the cylindrical coordinates formula use atan2(y, x) instead of arctan(y/x)?
The plain arctan function returns values only in the range (−π/2, π/2) and is undefined when x = 0. The atan2(y, x) function uses both the sign of x and the sign of y to determine the correct quadrant, returning values across the full (−π, π] range. For example, the point (−3, 0) gives atan2(0, −3) = π (pointing in the negative x-direction), but arctan(0/−3) = 0 — incorrect by π radians. This distinction matters for every point with x ≤ 0.
How do I convert cylindrical coordinates back to Cartesian coordinates?
To convert cylindrical (r, θ, z) back to Cartesian (x, y, z), use x = r cos θ, y = r sin θ, and keep z unchanged. For example, the cylindrical point (5, π/4, 3) converts to x = 5 cos(π/4) ≈ 3.536, y = 5 sin(π/4) ≈ 3.536, z = 3. Always ensure the angle θ is expressed in radians before applying the trigonometric functions, or convert from degrees using the relation θ_rad = θ_deg × π/180.
What is the difference between cylindrical and spherical coordinates?
Cylindrical coordinates (r, θ, z) measure the radial distance from the z-axis and leave z unchanged, whereas spherical coordinates (ρ, θ, φ) measure the radial distance from the origin and introduce a second angle φ (the polar or zenith angle from the z-axis). Cylindrical coordinates are preferable for problems with axial symmetry — such as cylinders, coils, and pipes — while spherical coordinates suit problems with central symmetry, such as gravitational or electrostatic fields radiating from a point source.
When should cylindrical coordinates be used instead of Cartesian coordinates?
Cylindrical coordinates simplify problems whenever the geometry or physics has rotational symmetry around a straight axis. Key use cases include evaluating triple integrals over cylinders or cones, describing electromagnetic fields around long straight wires and coaxial cables, modeling fluid flow in circular pipes, and performing axisymmetric structural analyses. In these contexts, using cylindrical coordinates reduces the number of variables a solution depends on, often converting partial differential equations into far simpler ordinary differential equations.
What is the volume element in cylindrical coordinates, and where does the factor r come from?
The volume element in cylindrical coordinates is dV = r dr dθ dz. The factor r — absent in the Cartesian element dV = dx dy dz — arises from the Jacobian determinant of the coordinate transformation: |∂(x,y,z)/∂(r,θ,z)| = r. As a practical check, integrating over a solid cylinder of radius R and height H gives ∫₀ᴴ ∫₀²π ∫₀ᴿ r dr dθ dz = πR²H, correctly recovering the standard cylinder volume formula. Forgetting the factor r is one of the most common errors in multivariable calculus.