Convert rectangular coordinates
Enter a three-dimensional point in the form (x, y, z). The calculator applies atan2(y, x) for reliable quadrant handling.
Example data
These examples show common rectangular points and their cylindrical forms. Angles below use the 0° to 360° convention.
| Rectangular point | Radius ρ | Angle θ | Cylindrical point |
|---|---|---|---|
| (3, 4, 8) | 5 | 53.13° | (5, 53.13°, 8) |
| (−3, 3, 2) | 4.24 | 135° | (4.24, 135°, 2) |
| (0, −6, 5) | 6 | 270° | (6, 270°, 5) |
| (7, 0, −1) | 7 | 0° | (7, 0°, −1) |
Understanding the Coordinate Systems
Rectangular coordinates locate a point with x, y, and z distances. Each distance follows a straight axis. Cylindrical coordinates describe that same point differently. They use a radius, an angle, and a height. The radius measures horizontal distance from the z-axis. The angle measures rotation around the z-axis. The height stays equal to the original z value. This method fits round objects naturally. It helps with pipes, wheels, circular paths, and rotational surfaces. Many users switch between them.
Why the Conversion Matters
A coordinate change does not move the point. It only changes the language used to describe it. Rectangular values are practical for boxes and straight edges. Cylindrical values are practical for circles and symmetry. A correct conversion can simplify an equation. It may reveal patterns. For example, a constant radius creates a cylinder. A constant angle creates a vertical half-plane. A constant height creates a horizontal plane. These ideas make three-dimensional graphs easier to interpret.
Formula Used
ρ = √(x² + y²)
θ = atan2(y, x)
z = z
The cylindrical radius is written as rho. Calculate it with rho = square root of x squared plus y squared. Find the angular coordinate with theta = atan2(y, x). The atan2 method checks both signs. It places the angle in the correct quadrant. The final cylindrical height is simply z. Therefore, the converted point is written as rho, theta, z. The calculator shows both degrees and radians. It also gives a normalized angle for easier comparison.
How to Use This Calculator
Enter the rectangular x coordinate in the first field. Enter the y coordinate in the next field. Enter the vertical z coordinate last. Choose degrees or radians for the primary answer. Select the angle range that you prefer. Choose a decimal precision for displayed values. Press Calculate Cylindrical Coordinates. The result appears above the form. Review the radius, angle, and unchanged height. Use the worked steps to verify each calculation. Download the result as a CSV file. You can also create a PDF summary.
Checking Your Answer
Start by checking the radius. It must equal the horizontal distance from the z-axis. A radius of zero means the angle is undefined. This happens when both x and y equal zero. Check the angle with the point quadrant. Positive x and positive y place it in quadrant one. Negative x and positive y place it in quadrant two. Negative x and negative y place it in quadrant three. Positive x and negative y place it in quadrant four. Finally, confirm that z did not change. This simple review prevents common conversion errors.
Practical Examples
Consider the rectangular point 3, 4, 8. Its radius is five because three squared plus four squared equals twenty-five. Its angle is about 53.13 degrees. The height remains eight. Now consider negative values. The point negative 3, 3, 2 has the same radius. Its angle becomes 135 degrees because it lies in quadrant two. This is why atan2 is safer than basic inverse tangent. It includes the coordinate signs and selects the correct direction.
Frequently asked questions
1. What are cylindrical coordinates?
Cylindrical coordinates identify a three-dimensional point with radius, angle, and height. They are written as rho, theta, z. The radius and angle describe the horizontal position. The z value describes the vertical position.
2. Which formula finds the cylindrical radius?
Use rho = square root of x squared plus y squared. This calculates the horizontal distance from the z-axis. The result cannot be negative. It becomes zero only when x and y are both zero.
3. Why does the calculator use atan2?
atan2 uses both y and x values. It determines the correct angle quadrant automatically. A basic inverse tangent can give an incomplete direction because it does not reliably distinguish every quadrant.
4. Does the z coordinate change?
No. The z coordinate remains exactly the same during this conversion. Both systems use the same vertical axis. Only the horizontal x and y description changes into radius and angle.
5. Can the angle be negative?
Yes. Negative angles are valid when using a signed range. This calculator can present angles from minus 180 degrees to 180 degrees. It can also normalize them from zero to 360 degrees.
6. What happens when x and y are zero?
The radius equals zero, so the point lies on the z-axis. Its angle is undefined. Any angle would identify the same horizontal location when the radius is zero.
7. Are degrees and radians both correct?
Yes. They express the same direction using different units. Degrees are often easier for visual interpretation. Radians are common in calculus, engineering formulas, and scientific computing.
8. How do I recognize the angle quadrant?
Look at the signs of x and y. Positive x and positive y form quadrant one. Continue counterclockwise for the remaining sign pairs. The calculator also labels the detected quadrant.
9. Can I use decimal coordinate values?
Yes. Enter integers, decimals, or negative numbers. The calculator accepts standard numeric values and lets you choose the number of decimal places shown in the result.
10. What does a constant radius represent?
A constant radius represents a circular cylinder around the z-axis. Every point stays the same horizontal distance from that axis. The angle and height may still change.
11. Can I save the conversion result?
Yes. After calculating, use Download CSV for spreadsheet data. Use Download PDF for a compact record. Both options are available directly in the result section.