Semicircle

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

$$L=2r+\pi r=r(2+\pi),$$ (1)

and the Area is

$$A=2\int_0^r \sqrt{r^2-y^2}\,dy={\textstyle{1\over 2}}\pi r^2.$$ (2)

The weighted mean of $y$ is

$$\left\langle{y}\right\rangle{}=2\int_0^r y\sqrt{r^2-y^2}\,dy={\textstyle{2\over 3}}r^3.$$ (3)

The Centroid is then given by

$$\bar y={\left\langle{y}\right\rangle{}\over A}={4r\over 3\pi}.$$ (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