prismatoidPrismatoid2007-06-04 17:38:162007-06-18 09:43:38Definitionaltitudebasesprismoidal formula% this is the default PlanetMath preamble. as your knowledge
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A \emph{prismatoid} is a polyhedron, possibly not convex, whose vertices all lie in one or the other
of two parallel planes.
The perpendicular distance between the two planes is called the \emph{altitude}
of the prismatoid.
The faces that lie in the parallel planes are called the \emph{bases}
of the prismatoid.
The \emph{midsection} is the polygon formed by cutting the prismatoid by
a plane parallel to the bases halfway between them.
The volume of a prismatoid is given by the \emph{prismoidal formula}:
$$
V = \frac{1}{6} h(B_1 + B_2 + 4M)
$$
where $h$ is the altitude, $B_1$ and $B_2$ are the areas of the bases and $M$
is the area of the midsection.
An alternate formula is :
$$
V = \frac{1}{4}h ( B_1 + 3S)
$$
where $S$ is the area of the polygon that is formed by cutting the prismatoid
by a plane parallel to the bases but 2/3 of the distance from $B_1$ to $B_2$.
A proof of the prismoidal formula for the case where
the prismatoid is convex is in \cite{Br}. It is also proved in \cite{Ha} for any prismatoid.
The alternate formula is proved in \cite{Ha}.
Some authors impose the condition that the lateral faces must be triangles
or trapezoids. However, this condition is unnecessary since it is easily shown
to hold.
\begin{thebibliography}{99}
\bibitem{Br}
A. Day Bradley, Prismatoid, Prismoid, Generalized Prismoid, \emph{The American Math. Monthly,}
\textbf{86}, (1979), 486-490.
\bibitem{Ha}
G.B. Halsted, \emph{Rational Geometry: A textbook for the Science of Space. Based on
Hilbert's Foundations}, second edition, John Wiley and Sons, New York, 1907
\end{thebibliography}