Cylindrical Coordinates Calculator
Convert between Cartesian (x, y, z) and cylindrical (r, θ, z) coordinate systems for three-dimensional geometry.
This free online cylindrical coordinates calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Cartesian x coordinate
Cartesian y coordinate
Cartesian z coordinate (unchanged in cylindrical)
Results
Radial Distance (r)
5
Angle θ (radians)
0.9273 rad
Angle θ (degrees)
53.13°
z Coordinate
5
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Cylindrical Coordinates Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Cylindrical Coordinates Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Cylindrical Coordinates Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Cylindrical Coordinates Calculator
Cylindrical coordinates provide an alternative way to specify points in three-dimensional space using a radial distance, an angle, and a height. This system extends polar coordinates into three dimensions by adding a z-axis. The radial distance r measures how far a point is from the z-axis, the angle θ (theta) measures the counterclockwise rotation from the positive x-axis in the xy-plane, and z gives the height above or below the xy-plane. Cylindrical coordinates are particularly useful for problems with rotational symmetry around a central axis, such as analyzing pipes, cylinders, electromagnets, and fluid flow in circular geometries. Engineers use cylindrical coordinates extensively in manufacturing when working with turned parts, drill holes, and any geometry that naturally revolves around an axis. This calculator converts Cartesian coordinates to their cylindrical equivalents, providing both radian and degree measures for the angle.
The Math Behind It
Formula Reference
Cartesian to Cylindrical
r = √(x² + y²), θ = atan2(y, x), z = z
Variables: x, y, z = Cartesian coordinates; r = radial distance; θ = azimuthal angle
Cylindrical to Cartesian
x = r·cos(θ), y = r·sin(θ), z = z
Variables: r = radial distance; θ = azimuthal angle; z = height
Worked Examples
Example 1: Convert a point from Cartesian to cylindrical
Convert the Cartesian point (3, 4, 5) to cylindrical coordinates.
The cylindrical coordinates are (5, 0.9273 rad, 5) or equivalently (5, 53.13°, 5).
Example 2: Point on the negative x-axis
Convert (-2, 0, 7) to cylindrical coordinates.
The cylindrical coordinates are (2, π rad, 7) or (2, 180°, 7).
Common Mistakes & Tips
- !Using arctan(y/x) instead of atan2(y, x), which fails to distinguish between opposite quadrants.
- !Forgetting that r is always non-negative — if you get a negative r, the angle needs adjustment.
- !Mixing up radians and degrees when interpreting the angle θ.
- !Neglecting that the z coordinate is identical in both systems — no conversion is needed for z.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
When should I use cylindrical coordinates instead of Cartesian?
Use cylindrical coordinates when your problem has symmetry around a central axis. Examples include analyzing flow through pipes, computing fields around wires, or integrating over cylindrical volumes. The symmetry simplifies the math considerably.
What is the range of the angle θ?
The angle θ typically ranges from -π to π when computed with atan2, or from 0 to 2π by convention. Both conventions describe the same set of points.
How do I convert back from cylindrical to Cartesian?
Use x = r·cos(θ) and y = r·sin(θ). The z coordinate remains the same. Make sure θ is in radians when using trigonometric functions.
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