Any horizontal
cross-section of a cloister vault is a square. This fact may be used to find the
volume of the vault using
Cavalieri's principle. Finding the volume in this way is often an exercise for first-year
calculus students, and was solved long ago by
Archimedes in Greece,
Zu Chongzhi in China, and
Piero della Francesca in Renaissance Italy; for more, see
Steinmetz solid. Assuming the intersecting barrel-vaults are semi-cylindrical, the volume of the vault is \frac{1}{3}s^3 where s is the length of the side of the square base. ==See also==