the surface area and volume of a sphere
A sphere is the set of all points a fixed distance r (the radius) from a centre — a perfectly round ball. Two numbers describe it: how much skin wraps around it (surface area) and how much space it fills (volume). Archimedes found both more than two thousand years ago and was so proud he asked for the result to be carved on his tomb.
The surface area is A = 4 · pi · r^2. A striking way to remember it: the surface area of a sphere equals exactly four times the area of one of its great circles (a great circle, area pi · r^2, is the largest circle you can draw on the sphere, like the equator). The volume is V = (4/3) · pi · r^3. Archimedes' jewel is that a sphere fills exactly two thirds of the smallest cylinder that encloses it (radius r, height 2r): that cylinder has volume pi · r^2 · 2r = 2·pi·r^3, and two thirds of that is (4/3)·pi·r^3.
Because volume grows as r^3 while surface area grows as r^2, big spheres have relatively little surface for their bulk — the square-cube law again. A common slip is to forget the 4 in the surface formula or the 4/3 in the volume formula; another is to use the diameter where the radius belongs, which inflates the volume eightfold. Both formulas can be derived cleanly by Cavalieri's principle, comparing a hemisphere to a cylinder with a cone removed.
A ball of radius 3 cm has surface area 4·pi·3^2 = 36·pi ≈ 113 cm^2 and volume (4/3)·pi·3^3 = 36·pi ≈ 113 cm^3 (the equal numbers here are a coincidence of r = 3, not a general rule).
A = 4·pi·r^2, V = (4/3)·pi·r^3; the sphere is two thirds of its enclosing cylinder.
Both formulas take the radius, not the diameter; using the diameter overstates the area fourfold and the volume eightfold, and the constants 4 and 4/3 are easy to drop by accident.