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

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

(1)

and the coordinates of the top polyhedron vertices are

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

(2)

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

R = [画像:1/2acsc(pi/(2n))]

(3)

r = [画像: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 = [画像:[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

[画像: [Rcos(pi/(2n))-r]^2+R^2sin^2(pi/(2n))+z^2=a^2 ]

(8)

[画像: z^2+R^2+r^2-2rRcos(pi/(2n))=a^2. ]

(9)