# Thread: Wierd reationship in spheres

1. ## 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.

2. The volume of the sphere is defined by summing infinitely many thin slices of the sphere. The radius of each disc or circle is$\displaystyle y=\sqrt{r^2-x^2}$ and the surface area of each circle is therefore $\displaystyle \pi y^2$.

For a sphere centered on the origin, we want the circles to be summed from -r to r so

$\displaystyle V=\int_{-r}^r \pi (r^2-x^2)dx = \frac{4}{3}\pi r^3$

3. Yes, that's true. But what does it have to do with the question?