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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),
\end{displaymath} (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),
\end{displaymath} (2)

where $R$ and $r$ are the Circumradii of the base and top
$\displaystyle R$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a\csc\left({\pi\over 2n}\right)$ (3)
$\displaystyle r$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a\csc\left({\pi\over n}\right),$ (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,
$\displaystyle {\bf b}$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}R\cos\left({\pi\over 2n}\right)\\  R\sin\left({\pi\over 2n}\right)\\  0\end{array}\right]$ (5)
$\displaystyle {\bf t}$ $\textstyle =$ $\displaystyle \left[\begin{array}{c}r\\  0\\  z\end{array}\right].$ (6)

Since all side lengths are $a$,
\vert{\bf b}-{\bf t}\vert^2=a^2.
\end{displaymath} (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
\end{displaymath} (8)

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

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

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


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

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© 1996-9 Eric W. Weisstein