A Cloister-vault, or domical vault, is a vault

Vault (architecture)

A Vault is an architectural term for an arched form used to provide a space with a ceiling or roof. The parts of a vault exert lateral thrust that require a counter resistance. When vaults are built underground, the ground gives all the resistance required...

 resulting from the intersection

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

 of two barrel-vaults crossing in a right angle.

Geometry

The intersection of the cloister vault with a horizontal plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 is a square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

. This fact may be used to find the volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

 of the vault using the slicing method of Integral calculus. Finding the volume in this way is often an exercise for first-year calculus students.

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.

Examples

• The chapel of the Portuguese Cardinal (James of Lusitania), in the Basilica di San Miniato al Monte
  Basilica di San Miniato al Monte
  San Miniato al Monte is a basilica in Florence, central Italy, standing atop one of the highest points in the city. It has been described as one of the finest Romanesque structures in Tuscany and one of the most beautiful churches in Italy. There is an adjoining Olivetan monastery, seen to the...
  
  , Florence
  Florence
  Florence is the capital city of the Italian region of Tuscany and of the province of Florence. It is the most populous city in Tuscany, with approximately 370,000 inhabitants, expanding to over 1.5 million in the metropolitan area....
  
  , Italy
  Italy
  Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
  
  
• Side chapels in the Church of St. Ippazio, Tiggiano
  Tiggiano
  Tiggiano is a town and comune in the Italian province of Lecce in the Apulia region of south-east Italy....
  
  , Italy
  Italy
  Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
  
  
• Maple Leaf Gardens
  Maple Leaf Gardens
  Maple Leaf Gardens is an indoor arena that was converted into a Loblawssupermarket and Ryerson University athletic centre in Toronto, on the northwest corner of Carlton Street and Church Street in Toronto's Garden District.One of the temples of hockey, it was home to the Toronto Maple Leafs of the...
  
   Roof
• Moscow Orphanage
  Moscow Orphanage
  The Moscow Orphanage or Foundling Home Vospitatel′nyj dom v Moskve was an ambitious project conceived by Catherine the Great and Ivan Betskoy, in the early 1760s...
  
  Roof of the Central Administrative Building