Cupola

An $n$-gonal cupola $Q_n$ (possible for only $n=3$, 4, 5) is a Polyhedron having $n$ Triangular and $n$ Square faces separating an $\{n\}$ and a $\{2n\}$ Regular Polygon. The coordinates of the base Vertices are

$$\left({R\cos\left[{\pi(2k+1)\over 2n}\right], R\sin\left[{\pi(2k+1)\over 2n}\right], 0}\right),$$ (1)

and the coordinates of the top Vertices are

$$\left({r\cos\left[{2k\pi\over n}\right], r\sin\left[{2k\pi\over n}\right], z}\right),$$ (2)

where $R$ and $r$ are the Circumradii of the base and top

$$R = {\textstyle{1\over 2}}a\csc\left({\pi\over 2n}\right)$$ (3)

$$r = {\textstyle{1\over 2}}a\csc\left({\pi\over n}\right),$$ (4)

and $z$ is the height, obtained by letting $k=0$ in the equations (1) and (2) to obtain the coordinates of neighboring bottom and top Vertices,

$${\bf b} = \left[\begin{array}{c}R\cos\left({\pi\over 2n}\right)\\ R\sin\left({\pi\over 2n}\right)\\ 0\end{array}\right]$$ (5)

$${\bf t} = \left[\begin{array}{c}r\\ 0\\ z\end{array}\right].$$ (6)

Since all side lengths are $a$,

$$\vert{\bf b}-{\bf t}\vert^2=a^2.$$ (7)

Solving for $z$ then gives

$$\left[{R\cos\left({\pi\over 2n}\right)-r}\right]^2+R^2\sin^2\left({\pi\over 2n}\right)+z^2=a^2$$ (8)

$$z^2+R^2+r^2-2rR\cos\left({\pi\over 2n}\right)=a^2$$ (9)

$$z = \sqrt{a^2-2rR\cos\left({\pi\over 2n}\right)-r^2-R^2}$$

$$= a\sqrt{1-{\textstyle{1\over 4}}\csc^2\left({\pi\over n}\right)}\,.$$ (10)

See also Bicupola, Elongated Cupola, Gyroelongated Cupola, Pentagonal Cupola, Square Cupola, Triangular Cupola

References

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