# Cylindrical Coordinates Calculator

URL: https://speedycalculator.com/calculators/math/cylindrical-coordinates/
Last updated: 2026-10-30

> The cylindrical coordinates calculator converts between Cartesian coordinates (x, y, z) and cylindrical coordinates (ρ, θ, z) in either direction.

## Inputs

- Conversion
- x
- y
- z
- Radius ρ
- Angle θ (°, rad, π rad, grad, turn)
- Height z

## Results

- Radius ρ
- Angle θ
- z
- x
- y
- z

## Formula

```
ρ = √(x² + y²),  θ = atan2(y, x),  z = z
x = ρ cos θ,  y = ρ sin θ
```

## Explanation

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

FieldWhat to enterConversionChoose one: Cartesian → cylindrical, Cylindrical → CartesianxA numberyA numberzA numberRadius ρA numberAngle θUnits: °, rad, π rad, grad, turnHeight zA 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:

xRadius ρ1.54.2722.254.5894353.755.48294.56.020867.2111

## 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.

## FAQ

**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.


## Sources

- [OpenStax – Calculus Volume 3](https://openstax.org/details/books/calculus-volume-3)
- [Wolfram MathWorld – Cylindrical coordinates](https://mathworld.wolfram.com/CylindricalCoordinates.html)