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Cylindrical Coordinates Calculator

Convert Cartesian (x, y, z) to cylindrical (r, θ, z).

r (radius) 5
θ (angle) (°) 53.13
z 5

Formula: r = √(x² + y²), θ = atan2(y, x)

Step-by-step with your numbers:
1. Values used:
2. x = 3
3. y = 4
4. z = 5
5.
6. r (radius) = 5
7. θ (angle) = 53.13°
8. z = 5
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Cylindrical coordinates describe a point by its distance from the z-axis, an angle, and a height.

How the Math Works

The cylindrical coordinates calculator transforms a point from Cartesian (x, y, z) to cylindrical (r, θ, z) coordinates using two fundamental formulas. The radial distance r is calculated as the square root of the sum of squared x and y values (r = √(x² + y²)), representing the distance from the z-axis to the point in the horizontal plane. The angular coordinate θ is determined using the arctangent function atan2(y, x), which returns the angle in radians between the positive x-axis and the point, accounting for the correct quadrant based on the signs of both x and y. The z-coordinate remains unchanged, as it represents the same height in both coordinate systems.

Practical Applications

This calculation is essential in engineering and physics when working with systems that exhibit cylindrical symmetry, such as analyzing electromagnetic fields around wires, fluid flow in pipes, or stress distributions in cylindrical tanks. Students use it to solve problems in multivariable calculus involving triple integrals with cylindrical coordinates, which simplifies calculations for symmetric objects. Engineers apply it when designing radar systems, antenna patterns, or any application where describing positions relative to a central axis is more natural than using rectangular coordinates.

Day-to-Day Use

While you may not calculate cylindrical coordinates manually every day, this mathematical transformation underlies technologies you encounter regularly. Your smartphone's GPS and mapping apps use similar coordinate transformations to represent locations, and navigation systems rely on polar and cylindrical coordinate concepts to calculate distances and bearings. Additionally, video game physics engines, robotics, and even the algorithms that help self-driving cars understand their position all incorporate these coordinate system conversions behind the scenes.

Worked example

(3, 4, 5) → r = 5, θ ≈ 53.13°, z = 5.

FAQ

How is this different from polar?

Cylindrical coordinates add the z (height) axis to 2D polar coordinates.

Related calculators

Spherical Coordinates Calculator Convert Cartesian (x, y, z) to spherical (ρ, θ, φ). Polar to Cartesian Converter (r,θ) → (x,y). Rectangular to Polar Coordinates (x,y) → (r,θ).