Difference between revisions of "Prismoid"
From Encyclopedia of Mathematics
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A [[Polyhedron|polyhedron]] two faces of which (its bases) are located in parallel planes, while the other faces are triangles or trapeziums, and, moreover, such that one side of each triangle face (that is not a base) and the two bases of each trapezium face (that is not a base) are sides of the bases of the prismoid (cf. Fig.). The volume of a prismoid is | A [[Polyhedron|polyhedron]] two faces of which (its bases) are located in parallel planes, while the other faces are triangles or trapeziums, and, moreover, such that one side of each triangle face (that is not a base) and the two bases of each trapezium face (that is not a base) are sides of the bases of the prismoid (cf. Fig.). The volume of a prismoid is | ||
| − | + | $$\frac h6(S+S'+4S''),$$ | |
| − | where | + | where $h$ is the distance between the bases, $S$ and $S'$ are their areas and $S''$ is the area of the intersection that has equal distance to both bases. |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p074840a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p074840a.gif" /> | ||
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====Comments==== | ====Comments==== | ||
| − | In | + | In $d$-space a prismoid is the convex hull of two $(d-1)$-polytopes lying in two distinct parallel hyperplanes. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Grünbaum, "Convex polytopes" , Wiley (1967)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Grünbaum, "Convex polytopes" , Wiley (1967)</TD></TR></table> | ||
Revision as of 16:30, 11 April 2014
A polyhedron two faces of which (its bases) are located in parallel planes, while the other faces are triangles or trapeziums, and, moreover, such that one side of each triangle face (that is not a base) and the two bases of each trapezium face (that is not a base) are sides of the bases of the prismoid (cf. Fig.). The volume of a prismoid is
$$\frac h6(S+S'+4S''),$$
where $h$ is the distance between the bases, $S$ and $S'$ are their areas and $S''$ is the area of the intersection that has equal distance to both bases.
Figure: p074840a
Comments
In $d$-space a prismoid is the convex hull of two $(d-1)$-polytopes lying in two distinct parallel hyperplanes.
References
| [a1] | B. Grünbaum, "Convex polytopes" , Wiley (1967) |
How to Cite This Entry:
Prismoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prismoid&oldid=31509
Prismoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prismoid&oldid=31509
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article