This is about relating differentials. Our volume is and our surface area .

Our volume and surface area do not change; this means and .

Now,

Thus

Dividing by the volumeabc, we get

(1)

Also:

Thus:

(2)

Solving (1) and (2) both fordcand combining:

Dividing by , we get

Similar solving of (1) and (2) fordaanddbgives the other equalities.

-Kevin C.