But why is a sphere’s surface area four times its shadow?

The first proof goes back to Greek times, due to Archimedes, who was charmed by the fact that a sphere has 2/3 the volume of a cylinder encompassing it, and 2/3 the surface area as well (if you consider the caps). Check out this video for another beautiful animation of that first proof: https://youtu.be/KZJw0AYn6_k

The formula is no mere coincidence.