📐 Cylindrical Coordinates Calculator
Convert between Cartesian and cylindrical coordinates
Cylindrical Coordinates (r, θ, z)
How to Use This Calculator
Select Conversion Direction
Choose whether to convert from cylindrical to Cartesian coordinates, or vice versa.
Enter Coordinates
Input the three coordinates based on your selection: either (r, θ, z) for cylindrical or (x, y, z) for Cartesian. Angle θ is in degrees.
Click Convert
Press "Convert" to get the equivalent coordinates in the other coordinate system.
Formula
Cylindrical → Cartesian:
x = r × cos(θ), y = r × sin(θ), z = z
Cartesian → Cylindrical:
r = √(x² + y²), θ = arctan2(y, x), z = z
Where:
- r = radial distance from z-axis (r ≥ 0)
- θ = azimuthal angle (0° to 360°)
- z = height along z-axis
- (x, y, z) = Cartesian coordinates
- arctan2(y, x) handles all quadrants correctly
Example: Convert cylindrical (r=2, θ=45°, z=3) to Cartesian
x = 2 × cos(45°) = 2 × 0.7071 ≈ 1.414
y = 2 × sin(45°) = 2 × 0.7071 ≈ 1.414
z = 3
Result: (1.414, 1.414, 3)
About Cylindrical Coordinates Calculator
The Cylindrical Coordinates Calculator converts between Cartesian (x, y, z) and cylindrical (r, θ, z) coordinate systems. Cylindrical coordinates extend polar coordinates to 3D by adding a z-axis component.
When to Use This Calculator
- Calculus: Convert coordinates for triple integrals and vector calculus
- Physics: Solve problems with cylindrical symmetry (cylinders, pipes, etc.)
- Engineering: Analyze rotational systems and cylindrical structures
- Mathematics: Work with problems that have cylindrical symmetry
- 3D Graphics: Convert between coordinate systems in rendering
- Electromagnetism: Solve Maxwell's equations in cylindrical coordinates
Why Use Our Calculator?
- ✅ Bidirectional Conversion: Convert in both directions seamlessly
- ✅ Angle Handling: Uses arctan2 for correct quadrant determination
- ✅ Step-by-Step Display: Shows the conversion formulas with your values
- ✅ Works with All Numbers: Handles decimals, fractions, and negatives
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding Cylindrical Coordinates
Cylindrical coordinates (r, θ, z) represent a point in 3D space using:
- r: Distance from the z-axis (radial distance, r ≥ 0)
- θ: Angle from positive x-axis in the xy-plane (azimuthal angle, 0° to 360°)
- z: Height along the z-axis (same as Cartesian z)
- Useful for problems with rotational symmetry around z-axis
- Similar to polar coordinates (r, θ) but extended to 3D with z
- Conversion preserves the z-coordinate (unchanged)
Real-World Applications
Electromagnetism: Electric and magnetic fields around wires and cylinders are naturally described in cylindrical coordinates due to rotational symmetry.
Fluid Dynamics: Flow through pipes and cylindrical channels uses cylindrical coordinates to simplify calculations.
Engineering: Rotating machinery, drill bits, and cylindrical structures are analyzed using cylindrical coordinates.
Frequently Asked Questions
What are cylindrical coordinates?
Cylindrical coordinates (r, θ, z) represent a point in 3D space using radial distance r from the z-axis, azimuthal angle θ in the xy-plane, and height z. They extend polar coordinates to 3D.
How do I convert from cylindrical to Cartesian?
Use: x = r × cos(θ), y = r × sin(θ), z = z. The z-coordinate stays the same. Make sure θ is in radians (or convert from degrees using θ × π/180).
How do I convert from Cartesian to cylindrical?
Use: r = √(x² + y²), θ = arctan2(y, x), z = z. Use arctan2 (not arctan) to get the correct angle in all quadrants.
Why use cylindrical coordinates instead of Cartesian?
Cylindrical coordinates simplify problems with rotational symmetry (around z-axis), such as cylinders, pipes, and circular objects. They make integrals and differential equations easier to solve.
Can r be negative?
Conventionally, r ≥ 0 (distance from z-axis). However, some texts allow negative r, which means going to the opposite side. This calculator requires r ≥ 0.
What's the difference from spherical coordinates?
Cylindrical uses (r, θ, z) where r is distance from z-axis. Spherical uses (ρ, θ, φ) where ρ is distance from origin and φ is polar angle. Cylindrical keeps one Cartesian axis (z).
How is this used in calculus?
Triple integrals, volume elements (dV = r dr dθ dz), and vector derivatives are often simpler in cylindrical coordinates for cylindrically symmetric problems.