Cylindrical Coordinates Calculator

Convert between cylindrical (r, θ, z) and Cartesian (x, y, z) coordinate systems with step-by-step solutions.

Select Conversion Direction

Result

Cartesian Coordinates
--
(x, y, z)
x --
y --
z --

Step-by-Step Solution

x = r cos(theta), y = r sin(theta), z = z

Understanding Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height component. A point in space is represented by three values: r (radial distance from the z-axis), theta (angle measured from the positive x-axis in the xy-plane), and z (height above the xy-plane). This system is particularly useful when dealing with problems that have rotational symmetry about an axis.

Conversion Formulas

Converting between cylindrical and Cartesian coordinates involves straightforward trigonometric relationships. These formulas are fundamental in calculus, physics, and engineering:

Conversion Reference

Cylindrical to Cartesian: x

The x-coordinate is found using the cosine of the angle.

x = r cos(theta)

Cylindrical to Cartesian: y

The y-coordinate is found using the sine of the angle.

y = r sin(theta)

Cartesian to Cylindrical: r

The radial distance is calculated from x and y.

r = sqrt(x^2 + y^2)

Cartesian to Cylindrical: theta

The angle is found using the arctangent function.

theta = atan2(y, x)

Height Component

The z-coordinate remains unchanged in both systems.

z = z (unchanged)

Degree-Radian Conversion

Convert between degrees and radians for the angle.

rad = deg x (pi / 180)

When to Use Cylindrical Coordinates

Cylindrical coordinates are the natural choice for problems involving:

  • Pipes and tubes: Flow analysis in cylindrical pipes, heat conduction in cylindrical shells.
  • Rotating machinery: Analysis of motors, turbines, and other rotating equipment.
  • Electromagnetic fields: Calculating fields around wires, solenoids, and cylindrical conductors.
  • Fluid dynamics: Vortex motion, flow around cylinders, and axisymmetric flows.
  • Antenna design: Radiation patterns and beam analysis for cylindrical antenna arrays.

Relationship to Other Coordinate Systems

Cylindrical coordinates sit between Cartesian (rectangular) and spherical coordinates in terms of complexity. While Cartesian coordinates use three perpendicular distances (x, y, z), cylindrical coordinates replace the x and y with a distance and angle (r, theta). Spherical coordinates go further by replacing all three Cartesian coordinates with a distance and two angles (rho, theta, phi).

Important Conventions

  • The radial distance r is always non-negative (r ≥ 0).
  • The angle theta is typically measured in radians (0 to 2pi) or degrees (0 to 360).
  • The z coordinate can be any real number, positive or negative.
  • When converting from Cartesian to cylindrical, use atan2(y, x) instead of atan(y/x) to get the correct quadrant.
  • This calculator accepts theta in degrees for convenience and shows the radian equivalent in the results.

Applications in Calculus

In multivariable calculus, cylindrical coordinates simplify triple integrals when the region of integration has cylindrical symmetry. The volume element in cylindrical coordinates is dV = r dr d(theta) dz, which includes the Jacobian factor r. This makes integration over cylindrical regions much more manageable compared to Cartesian coordinates.