Sphere: Surface and Volume

Significance

The result Archimedes asked to have carved on his tomb: a sphere has two-thirds the volume and two-thirds the surface of the cylinder that just contains it. The 4πr² is why every inverse-square law is inverse-square — light, gravity and electric flux spread over a spherical shell whose area grows as r². The r³ is why a planet's mass goes as the cube of its radius, and why its surface gravity, which is that mass over the radius squared, goes only as the first power of it: a world twice the size of another at the same density pulls twice as hard, not eight times.