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A prismatoid is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 where all vertices lie in two parallel planes. (If both planes have the same number of vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
,






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A prismatoid is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 where all vertices lie in two parallel planes. (If both planes have the same number of vertices
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
, 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


Pentagonal Pyramid
Pentagonal Prism
Pentagonal Antiprism
Pentagrammic Crossed Antiprism
Pentagonal Cupola
Pentagonal Frustum
Families of prismatoids include:

  • Pyramids
    Pyramid (geometry)

    In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
    , where one plane contains only a single point;
  • Wedges
    Wedge (geometry)

    In geometry, a wedge is a polyhedron solid defined by two triangles and three trapezoid faces. A wedge has five faces, nine edges, and six vertices....
    , where one plane contains only two points, or, the second plane contains two more points;
  • Prisms
    Prism (geometry)

    In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
    , where the polygons in each plane are congruent and joined by rectangles or parallelograms;
  • Antiprism
    Antiprism

    An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
    s, where the polygons in each plane are congruent and joined by an alternating strip of triangles;
  • crossed antiprisms;
  • Cupolas
    Cupola (geometry)

    In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of triangles and rectangles....
    , 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
    Frustum

    A frustum is the portion of a solid?normally a Cone or pyramid ?which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics to describe the 3d area which is visible on the screen ....
     obtained by truncation of a pyramid;
  • Quadrilateral
    Quadrilateral

    In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
    -faced hexahedral
    Hexahedron

    A hexahedron is a polyhedron with six faces. A Regular polyhedron hexahedron, with all its faces Square , is a cube.There are many kinds of hexahedra, some topologically similar to the cube and some not....
     prismatoids:
    1. Parallelepiped
      Parallelepiped

      In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
      s - six parallelogram
      Parallelogram

      In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
       faces
    2. Rhombohedron
      Rhombohedron

      In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombus. It is a special case of a parallelepiped where all edges are the same length....
      s - six rhombi
      Rhombus

      In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
       faces
    3. Trigonal trapezohedra
      Trigonal trapezohedron

      The trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms....
       - six congruent rhombi
      Rhombus

      In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length....
       faces
    4. Cuboid
      Cuboid

      In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
      s - six rectangular faces
    5. Quadrilateral frusta
      Frustum

      A frustum is the portion of a solid?normally a Cone or pyramid ?which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics to describe the 3d area which is visible on the screen ....
       - an apex
      Apex (geometry)

      In geometry, an apex is a descriptive label for a visual singular highest or most distant point or Vertex in an isosceles triangle, Pyramid or Cone , usually contrasting with the opposite side called the base....
      -truncated square pyramid
      Square pyramid

      In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
    6. Cube
      Cube

      A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
      s - six square faces


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