# Convert Spherical to Cylindrical Coordinates - Calculator

## Spherical and Rectangular Coordinates

Convert spherical to cylindrical coordinates using a calculator.
Using Fig.1 below, the trigonometric ratios and Pythagorean theorem, it can be shown that the relationships between spherical coordinates $(\rho,\theta,\phi)$ and cylindrical coordinates $(r,\theta,z)$ are as follows:
$r = \rho \sin \phi$ , $\theta = \theta$ , $z = \rho \cos \phi$       (I)
$\rho = \sqrt {r^2 + z^2}$ , $\theta = \theta$ , $\tan \phi = \dfrac{r}{z}$       (II)
with $0 \le \theta \lt 2\pi$ and $0 \le \phi \le \pi$ Fig.1 - Cylindrical and spherical coordinates The calculator calculates the cylindrical coordinates $r$, $\theta$ and $z$ given the spherical coordinates $\rho$ , $\theta$ and $\phi$ using the three formulas in I.

## Use Calculator to Convert Spherical to Cylindrical Coordinates

1 - Enter $\rho$ , $\theta$ and $\phi$, selecting the desired units for the angles, and press the button "Convert". You may also change the number of decimal places as needed; it has to be a positive integer.

 $\rho =$ 1 $\theta =$ 45 degrees radians $\phi =$ 45 degrees radians Number of Decimal Places = 5 $r =$ $\theta =$ (radians) $\theta =$ (degrees) $z =$