Convert between Cartesian (x, y, z) and cylindrical (r, θ, z) coordinates in 3D space. Enter your x, y, and z values to get the cylindrical radius, polar angle, and height — or go the other way by entering r, θ, and z to recover the Cartesian coordinates. Both conversion directions are supported, with results shown immediately.
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A cylindrical coordinate system is a three-dimensional coordinate system that specifies the position of a point using three values: r (the radial distance from the z-axis), θ (the polar angle in the xy-plane), and z (the height along the z-axis). It is essentially polar coordinates extended into 3D by adding the z-axis.
Given Cartesian coordinates (x, y, z), the cylindrical equivalents are: r = √(x² + y²), θ = atan2(y, x) (adjusted to be in [0°, 360°]), and z remains unchanged. The radius r is always non-negative.
Given cylindrical coordinates (r, θ, z), the Cartesian equivalents are: x = r·cos(θ), y = r·sin(θ), and z remains unchanged. Make sure θ is in the correct unit (degrees or radians) before applying the trigonometric functions.
Cylindrical coordinates (r, θ, z) use a radial distance from the z-axis and an angle in the xy-plane, keeping z as a linear height component. Spherical coordinates (ρ, θ, φ) use a radial distance from the origin and two angles — a polar angle and an azimuthal angle — to describe position. Cylindrical is best for problems with rotational symmetry about an axis, while spherical is ideal for problems with symmetry about a central point.
The angle θ (theta) is the polar angle measured in the xy-plane, starting from the positive x-axis and rotating counterclockwise. It ranges from 0° to 360° (or 0 to 2π radians) and indicates the angular position of the projection of the point onto the xy-plane.
By convention, r represents a physical distance from the z-axis, so it cannot be negative. The direction of the point relative to the origin is captured by the angle θ, which can range over a full 360°. This ensures each point has a unique representation.
Cylindrical coordinates are particularly useful when dealing with problems that have axial symmetry — for example, calculating volumes of cylinders, modeling fluid flow through pipes, or solving physics problems involving magnetic fields around a wire. They simplify the mathematics by reducing complex Cartesian expressions into more natural forms.
Yes, because the angle θ is periodic (repeating every 360° or 2π radians), a point can be represented with θ, θ + 360°, θ + 720°, etc. Conventionally, θ is restricted to [0°, 360°) to give a unique representation. The radius r is always kept non-negative.