| Version 6 |
Version 5 |
| A \emph{prismatoid} is a polyhedron, possibly not convex, whose vertices all lie in one or the other |
A \emph{prismatoid} is a polyhedron, possibly not convex, whose vertices all lie in one or the other |
| of two parallel planes. |
of two parallel planes. |
| The perpendicular distance between the two planes is called the \emph{altitude} |
The perpendicular distance between the two planes is called the \emph{altitude} |
| of the prismatoid. |
of the prismatoid. |
| The faces that lie in the parallel planes are called the \emph{bases} |
The faces that lie in the parallel planes are called the \emph{bases} |
| of the prismatoid. |
of the prismatoid. |
| The \emph{midsection} is the polygon formed by cutting the prismatoid by |
The \emph{midsection} is the polygon formed by cutting the prismatoid by |
| a plane parallel to the bases halfway between them. |
a plane parallel to the bases halfway between them. |
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The volume of a prismatoid is given by the \emph{prismoidal formula}:
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The volume of a prismatoid is given by the prismoidal formula:
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| $$ |
$$ |
| V = \frac{1}{6} h(B_1 + B_2 + 4M) |
V = \frac{1}{6} h(B_1 + B_2 + 4M) |
| $$ |
$$ |
| where $h$ is the altitude, $B_1$ and $B_1$ are the areas of the bases and $M$ |
where $h$ is the altitude, $B_1$ and $B_1$ are the areas of the bases and $M$ |
| is the area of the midsection. |
is the area of the midsection. |
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| An alternate formula is : |
An alternate formula is : |
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| $$ |
$$ |
| V = \frac{1}{4}h ( B_1 + 3S) |
V = \frac{1}{4}h ( B_1 + 3S) |
| $$ |
$$ |
| where $S$ is the area of the polygon that is formed by cutting the prismatoid |
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 $B_2$. |
by a plane parallel to the bases but 2/3 of the distance from $B_1$ to $B_2$. |
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| A proof of the prismoidal formula for the case where |
A proof of the prismoidal formula is in \cite{Br}. It is also proved in \cite{Ha}. |
| the prismatoid is convex is in \cite{Br}. It is also proved in \cite{Ha} for any prismatoid. |
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| The alternate formula is proved in \cite{Ha}. |
The alternate formula is proved in \cite{Ha}. |
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| Some authors impose the condition that the lateral faces must be triangles |
Some authors impose the condition that the lateral faces must be triangles |
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or trapezoids. However, this condition is unnecessary since it is easily shown
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or trapezoids. However, this condition is unnecessary since it is easily show |
| to hold. |
to hold. |
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| \begin{thebibliography}{99} |
\begin{thebibliography}{99} |
| \bibitem{Br} |
\bibitem{Br} |
| A. Day Bradley, Prismatoid, Prismoid, Generalized Prismoid, \emph{The American Math. Monthly,} |
A. Day Bradley, Prismatoid, Prismoid, Generalized Prismoid, \emph{The American Math. Monthly,} |
| \textbf{86}, (1979), 486-490. |
\textbf{86}, (1979), 486-490. |
| \bibitem{Ha} |
\bibitem{Ha} |
| G.B. Halsted, \emph{Rational Geometry: A textbook for the Science of Space. Based on |
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 |
Hilbert's Foundations}, second edition, John Wiley and Sons, New York, 1907 |
| \end{thebibliography} |
\end{thebibliography} |
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