Cylindrical coordinates are polar coordinates with a height added. They make problems with an axis of symmetry – pipes, coils, rotating fluids, electric fields around a wire – much simpler.
How to use the Cylindrical Coordinates Calculator
- Conversion
- x
- y
- z
- Radius ρ
- Angle θ
- Height z
Results update as you type, and everything is calculated in your browser.
How it works
The angle uses the two-argument arctangent so it lands in the correct quadrant, in the range −180° to 180°.
Inputs explained
| Field | What to enter |
|---|---|
| Conversion | Choose one: Cartesian → cylindrical, Cylindrical → Cartesian |
| x | A number |
| y | A number |
| z | A number |
| Radius ρ | A number |
| Angle θ | Units: °, rad, π rad, grad, turn |
| Height z | A number |
What each result means
- Radius ρ = sqrt([x]^2 + [y]^2)
- Angle θ = deg(atan2([y], [x]))
- x = [Radius ρ]×cos(rad([Angle θ]))
- y = [Radius ρ]×sin(rad([Angle θ]))
Example values
How the main result changes when x changes and everything else stays at the example values above:
| x | Radius ρ |
|---|---|
| 1.5 | 4.272 |
| 2.25 | 4.5894 |
| 3 | 5 |
| 3.75 | 5.4829 |
| 4.5 | 6.0208 |
| 6 | 7.2111 |
Formula
ρ = √(x² + y²), θ = atan2(y, x), z = z
x = ρ cos θ, y = ρ sin θ
Worked example
Inputs — Conversion: Cartesian → cylindrical; x: 3; y: 4; z: 5.
Results — Radius ρ = 5; Angle θ = 53.130102 °; z = 5.
Key terms
- Cylindrical coordinates
- (ρ, θ, z): polar coordinates plus height.
Frequently asked questions
How do cylindrical and spherical coordinates differ?
Cylindrical uses a height z; spherical uses a distance r and two angles.
What is θ for points on the z-axis?
Undefined (any angle works) because ρ = 0.
What is the volume element?
dV = ρ dρ dθ dz.
Can ρ be negative?
By convention no; use θ + 180° instead.