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'prismatoid'
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| Title of object: |
prismatoid |
| Canonical Name: |
Prismatoid |
| Type: |
Definition |
| Created on: |
2007-06-04 17:38:16 |
| Modified on: |
2007-06-04 17:40:50 |
| Classification: |
msc:51-00 |
| Defines: |
altitude, bases, prismoidal formula |
Preamble:
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Content:
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 prismoidal formula:
$$
V = \frac{1}{6} h(B_1 + B_2 + 4M)
$$
where $h$ is the altitude, $B_1$ and $B_1$ are the areas of the bases and $M$
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 $B_2$.
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