Calculator Form
Formula Used
Cylindrical coordinates represent a point as (r, θ, z), where r is the horizontal distance from the z-axis, θ is the angle measured from the positive x-axis, and z stays unchanged.
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
The 3D distance from the origin is √(x2 + y2 + z2), which is also √(r2 + z2).
How to Use This Calculator
- Choose the conversion mode.
- Enter either Cartesian values or cylindrical values.
- Select degrees or radians for the angle field.
- Choose the number of decimal places.
- Decide whether angles should be normalized.
- Press Convert Coordinates to view the result.
- Review the converted point, distance values, and graph.
- Use the CSV or PDF buttons to export the result.
Example Data Table
| Input Type | Input Values | Converted Output | Insight |
|---|---|---|---|
| Cartesian | (3, 4, 5) | (r = 5, θ = 53.1301°, z = 5) | A classic 3-4-5 planar triangle. |
| Cartesian | (-4, 4, 2) | (r = 5.6569, θ = 135°, z = 2) | The point lies in Quadrant II. |
| Cylindrical | (r = 6, θ = 210°, z = 7) | (x = -5.1962, y = -3.0000, z = 7) | The angle places the point in Quadrant III. |
| Cylindrical | (r = 8, θ = π/6, z = 1) | (x = 6.9282, y = 4.0000, z = 1) | This example uses a radian-based angle. |
FAQs
1. What are cylindrical coordinates?
Cylindrical coordinates describe a point with radius, angle, and height. They are written as (r, θ, z) and are useful for circular symmetry.
2. When should I use cylindrical instead of Cartesian coordinates?
Use cylindrical coordinates when problems involve circles, cylinders, rotations, or radial distance from an axis. They simplify many geometry and calculus calculations.
3. Why does z stay the same during conversion?
Both systems measure the same vertical position with z. Only the x and y plane changes representation between rectangular and polar-style form.
4. What does angle normalization do?
Normalization converts the angle into a standard interval, commonly 0 to 360 degrees or 0 to 2π radians. It keeps outputs easier to compare.
5. Can radius be negative?
A negative radius can represent the same point if the angle shifts by 180 degrees. This calculator converts negative radius inputs into a positive canonical form.
6. Why is atan2 used instead of arctan(y/x)?
atan2 detects the correct quadrant and handles axis cases more reliably. A simple arctan(y/x) can give ambiguous angles or fail when x equals zero.
7. What happens at the origin?
At the origin, the radius is zero and the angle is not unique. The calculator reports the angle as 0 by convention and adds a note.
8. What does the graph show?
The graph plots the Cartesian point in 3D space, shows its projection on the xy-plane, and draws guide lines to help you understand radius and height.