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 $$\pi$$r³, you get 4$$\pi$$r², 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$$\pi$$r², however, you get 8$$\pi$$r. My question is, what is the meaning of this 8$$\pi$$r? If you arrange the 4$$\pi$$r² into a new circle, say of radius p, the perimeter works out as 4$$\pi$$r, where 'r' is the old radius of the sphere. You would think that the 8$$\pi$$r, 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$$\pi$$r, 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 )