Cylindrical Coordinates Calculator
Convert between Cartesian and cylindrical coordinate systems
Convert Coordinate Systems
Cartesian Coordinates
Cylindrical Coordinates
Step-by-Step Calculation
Cartesian to Cylindrical Conversion
Given: Cartesian coordinates (x, y, z) = (3, 4, 5)
Formulas:
• ρ = √(x² + y²)
• θ = arctan(y/x) with proper quadrant handling
• z₂ = z₁
Calculations:
1. ρ = √(3² + 4²) = √(9 + 16) = 5.000000
2. θ = arctan(4/3) = 53.130102°
3. z = 5
Result: (ρ, θ, z) = (5.000000, 53.130102°, 5)
Example: Cartesian to Cylindrical Conversion
Given Point
Cartesian coordinates: (3, 4, 5)
Goal: Convert to cylindrical coordinates
Step-by-Step Solution
1. Calculate radial distance: ρ = √(3² + 4²) = √(9 + 16) = √25 = 5
2. Calculate angle: θ = arctan(4/3) = arctan(1.333...) ≈ 53.13°
3. Height remains the same: z = 5
4. Result: (ρ, θ, z) = (5, 53.13°, 5)
Coordinate Systems
Cartesian (x, y, z)
- • x: Distance along x-axis
- • y: Distance along y-axis
- • z: Distance along z-axis
Cylindrical (ρ, θ, z)
- • ρ (rho): Radial distance (≥ 0)
- • θ (theta): Azimuthal angle
- • z: Height (same as Cartesian)
Conversion Formulas
Cartesian → Cylindrical
Cylindrical → Cartesian
Constraints
Understanding Cylindrical Coordinates
What are Cylindrical Coordinates?
Cylindrical coordinates extend the 2D polar coordinate system to 3D space. This system is particularly useful for problems with cylindrical symmetry, such as pipes, towers, or rotating objects.
Coordinate Components
- •ρ (rho): Radial distance from z-axis (always ≥ 0)
- •θ (theta): Azimuthal angle measured from positive x-axis
- •z: Height along the z-axis (same as Cartesian)
Applications
Engineering
Fluid flow in pipes, heat transfer in cylinders
Physics
Electromagnetic fields, rotational mechanics
Mathematics
Triple integrals, surface area calculations
Computer Graphics
3D modeling, cylindrical projections
⚠️ Important Notes
- • ρ must be non-negative (≥ 0)
- • θ is typically in range (-π, π] or (-180°, 180°]
- • arctan(y/x) requires quadrant consideration
- • Use atan2(y, x) for proper quadrant handling