Enter Cartesian values
Formula used
Cartesian coordinates use distances along the x, y, and z axes. Cylindrical coordinates use radius, angle, and height.
θ = atan2(y, x)
z = z
x = r cos(θ), y = r sin(θ), z = z
For a function, replace every x with r*cos(theta) and every y with r*sin(theta). The z term stays unchanged.
How to use this calculator
- Enter the Cartesian x, y, and z values.
- Add a function if you need symbolic cylindrical substitution.
- Select the angle range and decimal precision.
- Press the calculate button to view r, θ, z, and function output.
- Check the quadrant note to confirm the angle direction.
Guide to Cartesian and Cylindrical Forms
Cartesian coordinates describe a point by three straight distances. The x value moves left or right. The y value moves forward or backward. The z value moves up or down. This system is simple for boxes, grids, and rectangular solids. Cylindrical coordinates describe the same point in a circular way. They use a radius, an angle, and a height. This makes them helpful when the shape, field, or motion rotates around an axis.
Why cylindrical form matters
Many real problems have circular symmetry. Pipes, tanks, wires, columns, coils, and rotating parts often fit this pattern. A Cartesian equation may look long in these cases. The same relation can become shorter after conversion. For example, x squared plus y squared becomes r squared. That simple change can make integration, graphing, and modeling much easier. It also helps students see the geometry behind a formula.
Point conversion idea
The radius is the horizontal distance from the z-axis. It is found from the x and y values only. The angle shows the direction of that horizontal projection. The height is still the original z value. The calculator uses atan2, not a plain tangent inverse. That is important. Atan2 reads the signs of both x and y. It places the angle in the correct quadrant.
Function conversion idea
A Cartesian function can also be rewritten. Replace x with r cos theta. Replace y with r sin theta. Keep z as z. Then simplify any common patterns. The most common pattern is x squared plus y squared. It becomes r squared because cosine squared plus sine squared equals one. This identity is the reason many circular formulas become clean in cylindrical coordinates.
Accuracy and angle choices
Angles may be shown in radians or degrees. Radians are common in calculus and advanced mathematics. Degrees are often easier to read in practical work. The same point can also have many equivalent angles. Adding 360 degrees, or two pi radians, points in the same direction. This calculator can normalize the angle to a positive range, or keep the signed atan2 range.
Where it is used
Use this form when rotation controls the problem. A heat map around a pipe, a magnetic field around a wire, or fluid motion inside a tank can be easier to express with r and theta. The distance from the axis becomes the main variable. The angle handles direction. The height handles vertical position. This split can reduce algebra, reveal symmetry, and prepare expressions for volume, surface, or line integrals in projects.
Practical checking
After conversion, check the result by returning to Cartesian form. Multiply r by cos theta to recover x. Multiply r by sin theta to recover y. The z value should match directly. Small differences can appear because of rounding. They are normal. Use more decimal places when the result will be used in engineering notes, code, or a multi-step calculation.
FAQs
What is a Cartesian to cylindrical conversion?
It changes a point from x, y, z form into r, θ, z form. The location stays the same. Only the coordinate description changes.
What does r mean in cylindrical coordinates?
r is the horizontal distance from the z-axis. It is calculated with the square root of x squared plus y squared.
What does theta mean?
Theta is the angle measured in the xy-plane from the positive x-axis. It shows the point direction around the z-axis.
Why is atan2 used?
Atan2 checks both x and y signs. This places theta in the correct quadrant, avoiding common inverse tangent mistakes.
Does the z value change?
No. In cylindrical coordinates, the height coordinate remains z. Only the x and y location becomes radius and angle.
How do I convert a function?
Replace x with r cos theta. Replace y with r sin theta. Keep z unchanged, then simplify the expression where possible.
What happens to x squared plus y squared?
It becomes r squared. This follows from x = r cos theta, y = r sin theta, and the identity cos²θ + sin²θ = 1.
Can theta be negative?
Yes. A signed angle may be negative in the atan2 range. The calculator can also normalize it between 0 and 2π.
Are radians or degrees better?
Radians are usually better for calculus and formulas. Degrees are easier for reading and reporting angles in many practical tasks.
Can one point have more than one theta?
Yes. Adding 2π radians or 360 degrees gives the same direction. Coordinate systems often choose one standard angle range.
Why should I check the quadrant?
The quadrant confirms the angle direction. It helps detect sign errors when x or y is negative or near zero.