In architecture, a cloister vault or domical vault is a vault with four concave surfaces (patches of cylinders) meeting at a point above the center of the vault.

It can be thought of as formed by two barrel vaults that cross at right angles to each other: the open space within the vault is the intersection of the space within the two barrel vaults, and the solid material that surrounds the vault is the union of the solid material surrounding the two barrel vaults. In this way it differs from a groin vault, which is also formed from two barrel vaults but in the opposite way: in a groin vault, the space is the union of the spaces of two barrel vaults, and the solid material is the intersection.

Geometry
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 and Zu Chongzhi in China; for more, see Steinmetz solid.

Assuming the intersecting barrel-vaults are semi-cylindrical, the volume of the vault is where s is the length of the side of the square base.

Characteristics
The barrel vault may be semicircular , low-arched or arched.

Arc-cloister vaults can be combined with barrel vaults to cover elongated rectangular surfaces.

The arch of cloister can be composed by the intersection of any number of vaults cradle. The base of the arch in the cloister can be a square in the simplest case or a regular polygon .

The dome of Florence’s Santa Maria del Fiore cathedral is an example of an arch-cloister vault built on an octagonal base .

When the barrel vault is semicircular, the surface of the arch in the cloister is twice the area of the base.

In the case where the base is a circle, one can assimilate it to a regular polygon with an infinity of sides. We can deduce that a dome with a semicircular profile, or hemispherical vault, is a special case of a semi-circular arch-cloister vault.

Examples
Side chapels in the Church of St. Ippazio, Tiggiano, Italy
Maple Leaf Gardens Roof
Moscow Orphanage Roof of the Central Administrative Building

Source From Wikipedia