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Vault


Vault

Let a vault consist of two equal half-cylinders of radius r 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

x^2+z^2=r^2
(1)
y^2+z^2=r^2
(2)

simultaneously gives

x=+/-sqrt(r^2-z^2)
(3)
y=+/-sqrt(r^2-z^2).
(4)

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

x=sqrt(r^2-z^2)
(5)
y=-usqrt(r^2-z^2)
(6)
z=z.
(7)

The surface area of the vault is therefore given by

 A=4intl(z)rdtheta,
(8)

where l(z) is the length of a cross section at height z and theta is the angle a point on the center of this line makes with the origin. But z=rsintheta, so

 dz=rcosthetadtheta=rsqrt(1-sin^2theta)dtheta=sqrt(r^2-z^2)dtheta,
(9)

and

 l(z)=2sqrt(r^2-x^2)
(10)
A=4int_0^r2rsqrt(r^2-z^2)(dz)/(sqrt(r^2-z^2))
(11)
=4int_0^r2rdz=8r^2.
(12)

The volume of the vault is

V=int_0^r(2sqrt(r^2-z^2))^2dz
(13)
=8/3r^3.
(14)

The geometric centroid is

 z^_=3/8r.
(15)

See also

Cylinder, Spherical Cap, Steinmetz Solid, Torispherical Dome

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References

Lines, L. Solid Geometry, with Chapters on Space-Lattices, Sphere-Packs, and Crystals. New York: Dover, pp. 112-113, 1965.Moore, M. "Symmetrical Intersections of Right Circular Cylinders." Math. Gaz. 58, 181-185, 1974.

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Vault

Cite this as:

Weisstein, Eric W. "Vault." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Vault.html

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