Hemisphere

Half of a Sphere cut by a Plane passing through its Center. A hemisphere of Radius can be given by the usual Spherical Coordinates

$\displaystyle x$	$\textstyle =$	$\displaystyle r\cos\theta\sin\phi$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle r\sin\theta\sin\phi$	(2)
$\displaystyle z$	$\textstyle =$	$\displaystyle r\cos\phi,$	(3)

where $\theta\in [0,2\pi)$ and $\phi\in [0, \pi/2]$ . All Cross-Sections passing through the

-axis are Semicircles.

The Volume of the hemisphere is

$\begin{displaymath} V=\pi \int_0^r (r^2-z^2)\,dz={\textstyle{2\over 3}}\pi r^3. \end{displaymath}$

(4)

The weighted mean of

over the hemisphere is

$\begin{displaymath} \left\langle{z}\right\rangle{}=\pi \int_0^r z(r^2-z^2)\,dz={\textstyle{1\over 4}}\pi r^2. \end{displaymath}$

(5)

The Centroid is then given by

$\begin{displaymath} \bar z={\left\langle{z}\right\rangle{}\over V}={\textstyle{3\over 8}}r \end{displaymath}$

(6)

(Beyer 1987).

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 133, 1987.