# Centroid of a semicircle vs. a semicircular arc

Why is the $y$ centroid of a semicircle and that of a semicircular arc different? Using Pappus' second theorem on a semicircle of radius $r$,

$\bar{y}=\frac{V}{2\pi A}=\frac{\frac{4}{3}\pi r^3}{2\pi (\frac{\pi r^2}{2})}=\frac{4r}{3\pi}$

Using Pappus' first theorem on a semicircular arc,

$\bar{y}=\frac{S}{2\pi s}=\frac{4\pi r^2}{2\pi \frac{2\pi r}{2}}=\frac{2r}{\pi}$

This is confirmed by Wikipedia. That page also claims that the area of the arc is $\pi r$. What does that even mean?

-
The half-disk, in its usual position, is more "bottom-heavy" than the wire half-circle. –  André Nicolas Apr 20 '13 at 8:29