On the Usenet newsgroup "sci.math", someone recently inquired why the area of a sphere is equal to 4 times the area of a disk of the same radius.
Using calculus, it is of course straightforward to prove the area formulas area(disk) = pi r^2 and area(sphere) = 4 pi r^2, whence the result follows.
But does anyone know a more direct way to see that
area(sphere) / area(disk) = 4 ???
A decomposition of a disk into a finite number of "nice" pieces that can be reassembled into a quarter-sphere seems out of the question, since the disk has Gaussian curvature = 0, while the sphere's curvature is 1/(r^2).
HOWEVER, maybe there is a convincing decomposition of the disk into an infinite number of "nice" pieces, each of which has 0 area, and which can be reassembled into a quarter sphere.
This method is probably fraught with pitfalls, but perhaps there is a valid approach like this which could somehow be made rigorous.
(For simplicity, let's assume from now on that r = 1.)