Calculator Input
Choose a conversion mode, enter known values, set angle units, and submit to calculate cylindrical or Cartesian coordinates.
Example Data Table
These sample conversions help verify expected behavior for common Cartesian and cylindrical coordinate inputs.
| Case | Input Type | Input Values | Output Values |
|---|---|---|---|
| Example 1 | Cartesian | (x, y, z) = (3, 4, 5) | (r, θ, z) = (5, 53.1301°, 5) |
| Example 2 | Cartesian | (x, y, z) = (-3, 3, 2) | (r, θ, z) = (4.2426, 135°, 2) |
| Example 3 | Cartesian | (x, y, z) = (0, -6, 7) | (r, θ, z) = (6, -90° or 270°, 7) |
| Example 4 | Cylindrical | (r, θ, z) = (8, 30°, 4) | (x, y, z) = (6.9282, 4, 4) |
| Example 5 | Cylindrical | (r, θ, z) = (5, 210°, -2) | (x, y, z) = (-4.3301, -2.5, -2) |
Formula Used
Cartesian to Cylindrical
r = √(x² + y²)
θ = atan2(y, x)
z = z
Cylindrical to Cartesian
x = r cos θ
y = r sin θ
z = z
Useful Derived Measures
Distance from origin = √(x² + y² + z²)
Cross-section circumference = 2πr
Cross-section disk area = πr²
Volume element in cylindrical form = r dr dθ dz
How to Use This Calculator
1. Select the conversion direction
Choose whether you want to convert Cartesian coordinates into cylindrical coordinates or convert cylindrical coordinates back into Cartesian form.
2. Enter the known coordinate values
Provide x, y, and z for Cartesian input, or provide r, θ, and z for cylindrical input. Use numeric values only.
3. Set angle unit and precision
Choose degrees or radians for the angular value. Then set the number of decimal places you want in the final result.
4. Normalize angle if needed
Enable normalization when you want angles reported inside the standard principal range instead of keeping raw angular output.
5. Submit and review outputs
Press the calculate button. The result appears above the form and shows converted coordinates, angle values, distance, and related geometric measures.
6. Export your result
Use the built-in CSV and PDF buttons to save the current result for reports, class notes, design checks, or later reference.
Frequently Asked Questions
1. What are cylindrical coordinates?
Cylindrical coordinates describe a point using radial distance r, azimuth angle θ, and height z. They are useful when symmetry exists around a central axis.
2. How is cylindrical form different from spherical form?
Cylindrical coordinates use one radius in the xy-plane plus height. Spherical coordinates use total radius and two angles to locate a point in space.
3. Why does this calculator use atan2(y, x)?
atan2 determines the correct angle quadrant by examining both x and y signs. This avoids ambiguity that appears when using a simple inverse tangent.
4. Can θ be negative?
Yes. Negative angles are valid and often represent clockwise rotation from the positive x-axis. You can also normalize them into a positive principal range.
5. Why must r be nonnegative?
The radial distance measures how far a point is from the z-axis, so it is defined as zero or positive. Negative values are not standard.
6. What happens when x and y are both zero?
The point lies on the z-axis. In that case, r becomes zero, while the azimuth angle is not unique because every angle points to the same axis.
7. When should I choose degrees instead of radians?
Use degrees for general classroom work or familiar geometric interpretation. Use radians for calculus, advanced mathematics, programming, and many scientific formulas.
8. What does the volume element r dr dθ dz mean?
It is the Jacobian factor used in integrals written in cylindrical coordinates. The extra r accounts for how area stretches as radius increases.