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)
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(2)
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simultaneously gives
(3)
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(4)
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One quarter of the vault can therefore be described by the parametric equations
(5)
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(6)
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(7)
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The surface area of the vault is therefore given by
(8)
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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)
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and
(10)
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(11)
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(12)
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The volume of the vault is
(13)
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(14)
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The geometric centroid is
(15)
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