Cylindrical Coordinates Calculator

The cylindrical coordinate system extends traditional polar coordinates into three dimensions by adding a height dimension (z). To convert from cylindrical coordinates (r, theta, z) to Cartesian coordinates (x, y, z), the calculator applies standard trigonometric relationships: x = r * cos(theta) and y = r * sin(theta), while the z coordinate remains unchanged. Conversely, when converting from Cartesian to cylindrical coordinates, the radial distance is found using the Pythagorean theorem, r = sqrt(x^2 + y^2), the azimuthal angle is computed using the arctangent function, theta = atan(y / x), and the height remains identical to the Cartesian z value.

Let us walk through a practical conversion example where we transform a set of cylindrical coordinates into Cartesian coordinates. Imagine a point in a cylindrical system defined by a radial distance r = 5, an azimuthal angle theta = 30 degrees (or approximately 0.5236 radians), and a vertical height z = 10. First, calculate the x-coordinate by multiplying the radius by the cosine of the angle: x = 5 * cos(30 degrees) = 5 * 0.866 = 4.33. Next, calculate the y-coordinate using the sine function: y = 5 * sin(30 degrees) = 5 * 0.5 = 2.5. The height z transfers directly over as 10. Thus, the resulting Cartesian coordinates for this point are (4.33, 2.5, 10).

Always verify whether your input angles are in degrees or radians before running calculations to prevent severe trigonometric discrepancies. When calculating the inverse tangent for theta, pay close attention to which quadrant your x and y values reside in, as standard arctangent functions can occasionally return angles restricted to specific quadrants. Finally, double-check your z-axis values, as they remain completely unaffected by conversions between Cartesian and cylindrical systems.