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\begin{figure}\begin{center}\BoxedEPSF{Semicircle.epsf scaled 1000}\end{center}\end{figure}

Half a Circle. The Perimeter of the semicircle of Radius $r$ is

L=2r+\pi r=r(2+\pi),
\end{displaymath} (1)

and the Area is
A=2\int_0^r \sqrt{r^2-y^2}\,dy={\textstyle{1\over 2}}\pi r^2.
\end{displaymath} (2)

The weighted mean of $y$ is
\left\langle{y}\right\rangle{}=2\int_0^r y\sqrt{r^2-y^2}\,dy={\textstyle{2\over 3}}r^3.
\end{displaymath} (3)

The Centroid is then given by
\bar y={\left\langle{y}\right\rangle{}\over A}={4r\over 3\pi}.
\end{displaymath} (4)

The semicircle is the Cross-Section of a Hemisphere for any Plane through the z-Axis.

See also Arbelos, Arc, Circle, Disk, Hemisphere, Lens, Right Angle, Salinon, Thales' Theorem, Yin-Yang

© 1996-9 Eric W. Weisstein