Vault

Let a vault consist of two equal half-cylinders of radius which intersect at right angles so that the lines of their intersections (the "groins") terminate in the polyhedron vertices of a square. Two vaults placed bottom-to-top form a Steinmetz solid on two cylinders.

Solving the equations

			(1)
			(2)

simultaneously gives

			(3)
			(4)

One quarter of the vault can therefore be described by the parametric equations

			(5)
			(6)
			(7)

The surface area of the vault is therefore given by

(8)

where is the length of a cross section at height and is the angle a point on the center of this line makes with the origin. But , so

(9)

and

(10)

			(11)
			(12)

The volume of the vault is

			(13)
			(14)

The geometric centroid is

(15)