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In [[geometry]], a '''prismatoid''' is a [[polyhedron]] where all [[vertex (geometry)|vertices]] lie in two parallel planes.  (If both planes have the same number of vertices,
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and the lateral faces are either [[parallelogram]]s or [[trapezoid]]s, it is called a '''prismoid'''.)
 
If the areas of the two parallel faces are A<sub>1</sub> and A<sub>3</sub>, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A<sub>2</sub>, and the height (the distance between the two parallel faces) is h, then the [[volume]] of the prismatoid is given by <math>V = \frac{h(A_1 + 4A_2 + A_3)}{6}</math> (This formula follows immediately by [[integral|integrating]] the area parallel to the two planes of vertices by [[Simpson's rule]], since that rule is exact for integration of [[polynomial]]s of degree up to 3, and in this case the area is at most a [[quadratic function]] in the height.)
 
== Prismatoid families ==
 
[[Image:Pentagonal pyramid.png|100px]]
[[Image:Pentagonal prism.png|100px]]
[[Image:Pentagonal_antiprism.png|100px]]
[[Image:Pentagrammic crossed antiprism.png|100px]]
[[Image:Pentagonal cupola.png|100px]]
[[Image:Pentagonal frustum.svg|100px]]
 
Families of prismatoids include:
 
* [[Pyramid (geometry)|Pyramids]], where one plane contains only a single point;
* [[Wedge (geometry)|Wedges]], where one plane contains only two points;
* [[Prism (geometry)|Prisms]], where the polygons in each plane are congruent and joined by rectangles or parallelograms;
* [[Antiprism]]s, where the polygons in each plane are congruent and joined by an alternating strip of triangles;
* [[Crossed antiprism]]s;
* [[cupola (geometry)|Cupolae]], where the polygon in one plane contains twice as many points as the other and is joined to it by alternating triangles and rectangles;
* [[Frustum|Frusta]] obtained by [[Truncation (geometry)|truncation]] of a pyramid;
* [[Quadrilateral]]-faced [[Hexahedron|hexahedral]] prismatoids:
*# [[Parallelepiped]]s - six [[parallelogram]] faces
*# [[Rhombohedron]]s - six [[rhombus]] faces
*# [[Trigonal trapezohedron|Trigonal trapezohedra]] - six congruent rhombus faces
*# [[Cuboid]]s - six rectangular faces
*# [[frustum|Quadrilateral frusta]] - an [[Apex (geometry)|apex]]-[[Truncation (geometry)|truncated]] [[square pyramid]]
*# [[Cube]]s - six square faces
 
== Higher dimensions ==
 
In general a [[polytope]] is prismatoidal if its vertices exist in two [[hyperplane]]s. For example in 4-dimension, two polyhedra can be placed in 2 parallel 3-spaces, and connected with polyhedral sides.
 
[[File:4D Tetrahedral Cupola-perspective-cuboctahedron-first.png|320px]]<BR>A tetrahedral-cuboctahedral cupola.
 
==External links==
* {{MathWorld | urlname=Prismatoid | title=Prismatoid}}
 
{{Polyhedron navigator}}
 
{{polyhedron-stub}}
[[Category:Polyhedra]]
[[Category:Prismatoid polyhedra]]

Latest revision as of 06:27, 13 January 2015

Every respectable home working job may a spot of effort to get off the ground, so do not expect an income from 1. But are generally three basic still some easy home working jobs to test out.

When I started my online business career, I experienced similar things. Everytime I obtained "how-to" product, I still existed with more questions than answers; it seemd coordinator . never ending loop info overload. In fact, it was so overwhelming that I completely stopped trying to pursue enterprise opportunities above a decade.

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