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Cupola

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An n-gonal cupola Q_n is a polyhedron having n obliquely oriented triangular and n rectangular faces separating an {n} and a {2n} regular polygon, each oriented horizontally. The coordinates of the base polyhedron vertices are

(Rcos[(pi(2k+1))/(2n)],Rsin[(pi(2k+1))/(2n)],0),
(1)

and the coordinates of the top polyhedron vertices are

(rcos[(2kpi)/n],rsin[(2kpi)/n],z),
(2)

where R and r are the circumradii of the base and top

R=1/2acsc(pi/(2n))
(3)
r=1/2acsc(pi/n),
(4)

and z is the height.

Cupola3
Cupola4
Cupola5

A cupola with all unit edge lengths (in which case the triangles become unit equilateral triangles and the rectangles become unit squares) is possible only for n=3, 4, 5, in which case the height z can be obtained by letting k=0 in the equations (1) and (2) to obtain the coordinates of neighboring bottom and top polyhedron vertices,

b=[Rcos(pi/(2n)); Rsin(pi/(2n)); 0]
(5)
t=[r; 0; z].
(6)

Since all side lengths are a,

|b-t|^2=a^2.
(7)

Solving for z then gives

[Rcos(pi/(2n))-r]^2+R^2sin^2(pi/(2n))+z^2=a^2
(8)
z^2+R^2+r^2-2rRcos(pi/(2n))=a^2.
(9)

Solving for z then gives

z=sqrt(a^2-2rRcos(pi/(2n))-r^2-R^2)
(10)
=asqrt(1-1/4csc^2(pi/n)).
(11)

See also

Bicupola, Elongated Cupola, Gyroelongated Cupola, Pentagonal Cupola, Rotunda, Square Cupola, Triangular Cupola

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References

Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169-200, 1966.

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Cupola

Cite this as:

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

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