Changing Coordinate Systems: Rectangular and Cylindrical

Rectangular Coordinates (Cartesian Coordinates) Cylindrical Coordinates

Since the z coordinate is the same in both coordinate systems, we just need to relate x and y to r and θ. We have the following triangles on the xy plane:

Rectangular Coordinates (Cartesian Coordinates) Cylindrical Coordinates

Rectangular Coordinates on xy Plane Cylindrical Coordinates on xy Plane

Comparing these we see that

x = r cos θ
y = r sin θ
r = sqrt(x² + y²)
θ = tan^-1(y/x)

Back to Describing Surfaces Using Different Coordinate Systems.

Rectangular Coordinates (Cartesian Coordinates)	Cylindrical Coordinates