Cartesian to Cylindrical Coordinate Converter
Convert Cartesian coordinates (x, y, z) to Cylindrical coordinates (ρ, φ, z) instantly. Ideal for students, engineers, and professionals working with 3D coordinate systems.
Formulas
\[ \rho = \sqrt{x^2 + y^2} \]
\[ \phi = \arctan\left(\frac{y}{x}\right) \]
\[ z = z \]
How It Works
The cylindrical coordinate system extends the two-dimensional polar coordinate system to three dimensions by adding a height (z) value. The conversion formulas transform Cartesian coordinates to their cylindrical equivalents.
Example Calculation
Given Cartesian coordinates (3, 4, 5):
- ρ = √(3² + 4²) = √(9 + 16) = √25 = 5
- φ = arctan(4/3) ≈ 53.13°
- z = 5
Result: Cylindrical coordinates are (5, 53.13°, 5)
Glossary of Terms
- ρ (rho): The radial distance from the z-axis (always non-negative)
- φ (phi): The azimuthal angle in the xy-plane from the positive x-axis
- z: The height above the xy-plane (same as Cartesian z)
- Cartesian coordinates: A coordinate system using (x, y, z) to specify positions in 3D space
- Cylindrical coordinates: A coordinate system using (ρ, φ, z) ideal for problems with cylindrical symmetry
Frequently Asked Questions
What is the cylindrical coordinate system?
The cylindrical coordinate system extends 2D polar coordinates to 3D by adding a height (z) value. It's particularly useful for problems involving cylinders, spirals, and geometries with axial symmetry.
Why use cylindrical coordinates?
Cylindrical coordinates simplify calculations for problems with cylindrical symmetry, such as fluid flow in pipes, electromagnetic fields around wires, or motion along helical paths.
How do I convert from cylindrical back to Cartesian coordinates?
Use the reverse formulas: x = ρ cos(φ), y = ρ sin(φ), and z = z.
Data Source and Methodology
All calculations follow standard mathematical transformation formulas for coordinate system conversions, recognized in mathematics, physics, and engineering fields.