Cloister vault

Domical vaults can have other polygons as cross-sections (especially octagons) rather than being limited to squares.

[3] 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,[4] and was solved long ago by Archimedes in Greece, Zu Chongzhi in China, and Piero della Francesca in Renaissance Italy;[5] for more, see Steinmetz solid.

Assuming the intersecting barrel-vaults are semi-cylindrical, the volume of the vault is

Cloister vault
The squared dome of the Great Synagogue of Rome
A dome tent shaped as a cloister vault
The central vault of the Patuxai monument in Laos