Math July 13, 2026 · 8 Min Read

Cylindrical Coordinates Calculator – Guide & Formulas

Convert between cylindrical and Cartesian coordinates instantly. Enter (r, θ, z) or (x, y, z) to see the conversion with step-by-step trigonometric breakdown.

Convert between cylindrical and Cartesian coordinates with our free online calculator. Enter (r, θ, z) or (x, y, z) to see the conversion with step-by-step breakdown.

Key Takeaway

Use the free Cylindrical Coordinates Calculator to convert between cylindrical and cartesian coordinates instantly. enter (r, θ, z) or (x, y, z) to see the conversion with step-by-step trigonometric breakdown. Get instant results with step-by-step explanations.

How to Use the Cylindrical Coordinates Calculator

  1. Step 1: Select conversion direction: Cylindrical to Cartesian or Cartesian to Cylindrical
  2. Step 2: Enter the coordinates in the selected system
  3. Step 3: Choose angle units: degrees or radians
  4. Step 4: Click Calculate to see the converted coordinates
  5. Step 5: Review the trigonometric breakdown and step-by-step computation

The Formula

Cylindrical to Cartesian: x = r×cos(θ), y = r×sin(θ), z = z. Cartesian to Cylindrical: r = √(x²+y²), θ = atan2(y,x), z = z.

Variable Definitions

  • r: Radial distance from the z-axis (always non-negative)
  • θ: Azimuthal angle from the positive x-axis (0 to 2π or 0° to 360°)
  • z: Height along the z-axis (same in both systems)
  • x: Cartesian x-coordinate
  • y: Cartesian y-coordinate
  • atan2(y,x): Two-argument arctangent that gives the correct quadrant for θ

Convert Cylindrical (3, 60°, 5) to Cartesian

Convert r = 3, θ = 60°, z = 5 to Cartesian coordinates.

  1. Step 1: Identify cylindrical coordinates: r = 3, θ = 60°, z = 5.
  2. Step 2: Convert angle to radians: 60° = π/3 ≈ 1.0472 rad.
  3. Step 3: x = r × cos(θ) = 3 × cos(60°) = 3 × 0.5 = 1.5.
  4. Step 4: y = r × sin(θ) = 3 × sin(60°) = 3 × 0.866 = 2.598.
  5. Step 5: z = 5 (unchanged). Cartesian coordinates: (1.5, 2.598, 5).

Frequently Asked Questions

What are cylindrical coordinates?

Cylindrical coordinates (r, θ, z) describe a point using its distance from the z-axis (r), angle from the x-axis (θ), and height (z). They are an extension of polar coordinates to 3D.

How are cylindrical coordinates different from polar coordinates?

Cylindrical coordinates add a z-axis to polar coordinates. Polar coordinates are 2D (r, θ), while cylindrical coordinates are 3D (r, θ, z). The z-value is the same in both cylindrical and Cartesian.

When should I use cylindrical coordinates?

Cylindrical coordinates are ideal for problems with rotational symmetry about an axis, such as pipes, cylinders, coils, magnetic fields around wires, and volume integrals of cylindrical objects.

How do I convert back from Cartesian to cylindrical?

Use r = √(x² + y²), θ = atan2(y, x) (which handles all four quadrants correctly), and z = z. The z-coordinate stays the same in both systems.

What is atan2(y,x)?

atan2 is the two-argument arctangent function that returns the correct angle for all four quadrants. Unlike simple arctan(y/x), it handles negative values and the case where x = 0 correctly.

Can r be negative in cylindrical coordinates?

By convention, r is non-negative (r ≥ 0). If a negative r appears, it can be converted by adding π to θ and taking r positive. The calculator uses the standard convention.