What is the Meaning of 8πr in Differentiating Sphere Volume?

[Aeneas]

If you differentiate the formula for the volume of a solid sphere, 4/3 \pir³, you get 4\pir², the formula for the surface area. This seems to make sense, as, onion-like, the sphere is made up of successive surface areas, so the rate of change will be the surface area for any given r. If you differentiate 4\pir², however, you get 8\pir. My question is, what is the meaning of this 8\pir? If you arrange the 4\pir² into a new circle, say of radius p, the perimeter works out as 4\pir, where 'r' is the old radius of the sphere. You would think that the 8\pir, then, must be the perimeter of the 'net' of the sphere, whatever that is. Is that this '9-gore' idea, or is that just a practical approximation? Is there a simple formula for its perimeter? If it is 8\pir, is there any significance in the fact that it seems to be twice the smallest perimeter that would enclose such an area?- or is my reasoning all wrong? Sorry about the flying pies!

Thanks in anticipation,
Aeneas

 

[tiny-tim]

HI Aeneas!

(have a pi: π )

The reason why differentiating the volume gives you the surface area is that the volume is made up of shells of that surface area.

(Similary, differentiating πr² for the area of a circle gives you 2πr for the circumference, and it also works for the volume and surface area of any n-dimensional "sphere".)

But the surface area isn't made up of shells of anything, so differentiating it doesn't give anything geometric.

(sorry, but the rest of your post I didn't follow )