A thin membrane covers the surface of the (spherical) earth.

geometry/cover.earth

Problem

A thin membrane covers the surface of the (spherical) earth.  One
square meter is added to the area of this membrane to form a larger
sphere.  How much is added to the radius and volume of this membrane?

Show solution

We know that V = (4/3)*pi*r^3 and A = 4*pi*r^2.
We need to find out how much V increases if A increases by 1 m^2.

  dV / dr = 4 * pi * r^2
  dA / dr = 8 * pi * r
  dV / dA = (dV / dr) / (dA / dr)
          = (4 * pi * r^2) / (8 * pi * r)
          = r/2
          = 3,250,000 m

If the area of the cover is increased by 1 square meter,
then the volume it contains is increased by about 3.25 million cubic meters.

We seem to be getting a lot of mileage out of such a small square of cotton.
However, the new cover would not be very high above the surface of the
planet -- about 6 nanometers (calculate dr/dA).