Hemisphere Volume

Half a sphere holds half a sphere's worth, so $V = \ frac{2}{3}\pi r^3$. What makes it more than a trivial halving is the fact Archimedes proved and asked to have engraved on his tombstone: a sphere occupies exactly two thirds of the cylinder that just contains it. Cut everything in half and the hemisphere occupies two thirds of its enclosing cylinder too, which is where the $\ frac{2}{3}$ in front of $\pi r^3$ literally comes from.

Practically it is the bowl formula and the dome formula. A mixing bowl 30 cm across holds $\ frac{2}{3}\pi(0.15)^3 \approx 7.07$ litres brim full, which is why a "10 litre" bowl is larger than most people expect. It is also the capacity of a hemispherical tank head, the displacement of a dome-topped piston crown, and the earth volume in a hemispherical excavation.