In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be or . If both planes have the same number of vertices, and the lateral faces are either or trapezoids, it is called a prismoid.
Volume
If the areas of the two parallel faces are and , the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is , and the height (the distance between the two parallel faces) is , then the
volume of the prismatoid is given by
This formula follows immediately by
integral the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of
of degree up to 3, and in this case the area is at most a quadratic function in the height.
Prismatoid families
Families of prismatoids include:
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Pyramids, in which one plane contains only a single point;
[.]
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Wedges, in which one plane contains only two points;
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Prisms, whose polygons in each plane are congruent and joined by rectangles or parallelograms;
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, whose polygons in each plane are congruent and joined by an alternating strip of triangles;
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;
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Cupolae, in which the polygon in one plane contains twice as many points as the other and is joined to it by alternating triangles and rectangles;
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Frustum obtained by truncation of a pyramid or a cone;
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Quadrilateral-faced hexahedron prismatoids:
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– six parallelogram faces
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– six rhombus faces
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Trigonal trapezohedra – six congruent rhombus faces
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– six rectangular faces
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frustum – an apex-truncated square pyramid
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Cube – six square faces
Higher dimensions
In general, a
polytope is prismatoidal if its vertices exist in two
. For example, in four dimensions, two polyhedra can be placed in two parallel 3-spaces, and connected with polyhedral sides.
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