prismatoid

In geometry, a prismatoid is a polyhedron where all vertices lie in two parallel planes. (If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid.)

If the areas of the two parallel faces are A₁ and A₃, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A₂, and the height (the distance between the two parallel faces) is h, then the volume of the prismatoid is given by V = h(A₁ + 4A₂ + A₃)/6.

Prismatoid families

Families of prismatoids include:

• Pyramids, where one plane contains only a single point;
• Wedges, where one plane contains only two points, or, the second plane contains two more points;
• Prisms, where the polygons in each plane are congruent and joined by rectangles or parallelograms;
• Antiprisms, where the polygons in each plane are congruent and joined by an alternating strip of triangles;
• Crossed antiprisms;
• Cupolas, where the polygon in one plane contains twice as many points as the other and is joined to it by alternating triangles and rectangles;
• Frusta obtained by truncation of a pyramid;
• Quadrilateral-faced hexahedral prismatoids:
  1. Parallelepipeds - six parallelogram faces
  2. Rhombohedrons - six rhombus faces
  3. Trigonal trapezohedra - six congruent rhombus faces
  4. Cuboids - six rectangular faces
  5. Quadrilateral frusta - an apex-truncated square pyramid
  6. Cubes - six square faces

External links

• Weisstein, Eric W., "Prismatoid" from MathWorld.

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