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Half of a Sphere cut by a Plane passing through its Center. A hemisphere of Radius $r$ 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 $z$-axis are Semicircles.

The Volume of the hemisphere is

V=\pi \int_0^r (r^2-z^2)\,dz={\textstyle{2\over 3}}\pi r^3.
\end{displaymath} (4)

The weighted mean of $z$ over the hemisphere is
\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
\bar z={\left\langle{z}\right\rangle{}\over V}={\textstyle{3\over 8}}r
\end{displaymath} (6)

(Beyer 1987).

See also Semicircle, Sphere


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

© 1996-9 Eric W. Weisstein