# prismatoid

A *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

*altitude*of the prismatoid. The faces that lie in the parallel planes are called the

*bases*of the prismatoid. The

*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 ${B}_{2}$ 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 [1]. It is also proved in [2] for any prismatoid. The alternate formula is proved in [2].

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.

## References

Title | prismatoid |

Canonical name | Prismatoid |

Date of creation | 2013-03-22 17:12:03 |

Last modified on | 2013-03-22 17:12:03 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 10 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 51-00 |

Related topic | SimpsonsRule |

Related topic | Volume2 |

Related topic | TruncatedCone |

Defines | altitude |

Defines | bases |

Defines | prismoidal formula |