Rectangular to Cylindrical Coordinates Calculator

Calculator Input

Coordinate Graph

The graph shows the point, its projection on the xy-plane, and the radial path from the origin.

Formula Used

The rectangular point is written as (x, y, z). The cylindrical point is written as (r, θ, z).

• Radius: r = √(x² + y²)
• Angle: θ = atan2(y, x)
• Height: z = z
• Inverse check: x = r cos(θ) and y = r sin(θ)
• 3D distance: d = √(x² + y² + z²)

The calculator uses atan2(y, x) because it returns the correct quadrant for positive and negative coordinate signs.

How to Use This Calculator

1. Enter the rectangular x, y, and z coordinate values.
2. Choose the angle range you want to use.
3. Select the preferred angle output format.
4. Set decimal places for rounding.
5. Add a unit label if your values use meters, feet, inches, or another unit.
6. Press the convert button to view the result above the form.
7. Use the graph to check direction and height visually.
8. Download the result as CSV or PDF for records.

Example Data Table

Article

Understanding Coordinate Conversion

Rectangular coordinates describe a point with x, y, and z distances. Cylindrical coordinates describe the same point with radius, angle, and height. The radius shows distance from the vertical axis. The angle shows direction around that axis. The height is the original z value.

Why Cylindrical Form Helps

Cylindrical form is useful when shapes, forces, fields, pipes, tanks, and paths rotate around an axis. Many engineering and science problems become simpler after conversion. A circular boundary can be described with one radius value. A rotating motion can be tracked with one angle. This reduces clutter and improves interpretation.

Reading the Results

This calculator reports r, theta, and z. It also gives theta in degrees and radians. The quadrant note helps confirm the angle direction. The inverse check rebuilds x and y from the cylindrical values. Small differences can appear because of rounding. Use more decimal places when you need tighter reporting.

Working With Signs

The atan2 method handles positive and negative x and y values. That means the correct quadrant is found automatically. When x and y are both zero, the radius is zero. In that case, the direction is not unique. The tool explains that special case so the answer stays clear.

Practical Uses

Students can use the page for homework checking. Teachers can create examples for lessons. Surveying, robotics, graphics, fluid systems, and electromagnetic work can also benefit. The table and graph make the result easier to understand. Export buttons help store the calculation in a spreadsheet or a report.

Good Input Habits

Always keep units consistent. Coordinates may be meters, feet, inches, or any linear unit. The angle is unitless, but it can be shown in degrees or radians. Check the selected angle range before copying the result. Use the example table to compare known points with your own values. Enter precise values when the calculation supports design choices. Review the graph to catch sign mistakes.

Accuracy Notes

For most classroom problems, four or six decimal places are enough. For fabrication or simulation, use a higher setting. Keep the original numbers with the exported file, because rounded displays can hide small changes during review.

FAQs