Wierd reationship in spheres

I found a strange relationship between a sphere's volume and area, and a circle's area and circumference.

For spheres:

$\displaystyle \frac{d}{dr}V=\frac{d}{dr}\frac{4}{3}\pi r^3=4\pi r^2=A$

Where V is the volume and A is the area.

For cirles:

$\displaystyle \frac{d}{dr}A=\frac{d}{dr}\pi r^2=2\pi r=O$

Where A is the area and O is the circumference.

I'm sure that this applies to higher dimensions as well.

What gives rise to this relation ship? Is it a "coincidence"?

Thanks in advance.