Polyhedron

A solid whose boundary consists of a finite number of polygonal faces, that is, planar regions that are bounded by polygons. The sides of the faces are edges of the polyhedron; the vertices of the faces also are vertices of the polyhedron.


Most polyhedra met in applied geometry are convex and simply connected. A polyhedron is convex if it passes this test: if any face is placed coincident with a plane, then all other points of the polyhedron lie on the same side of that plane. A more informal test is to imagine enclosing the polyhedron within a stretched elastic membrane; the polyhedron is convex if all points on the boundary are in contact with the membrane. A simply connected polyhedron has a boundary that is topologically equivalent to a sphere: if the boundary were made of some perfectly elastic material, then the boundary could be distorted into a sphere without tearing or piercing the surface. A simply connected polyhedron is said to be eulerian, because the number of faces F, the number of edges E, and the number of vertices V satisfy Euler's formula 


The remainder of this article will deal only with simply connected convex polyhedra.


A polyhedron has a volume, a measure of the amount of space enclosed by the boundary surface. As noted in cases below, often the volume can be determined if certain other measurements associated with the polyhedron are known.


The fewest number of faces that a polyhedron might have is four. “Poly” signifies “many,” but since an n-hedron (a polyhedron having n faces) can be transformed into an n + 1-hedron by truncating the polyhedron near a vertex, polyhedra exist having any number of faces greater than three. Some polyhedra have names that convey the number of faces (but not the shape) of the polyhedron: tetrahedron, 4 faces; pentahedron, 5 faces; hexahedron, 6 faces; octahedron, 8 faces; dodecahedron, 12 faces; and icosahedron, 20 faces.
Many polyhedra have properties that allow them to be placed in one or more of the categories below.


Prismatoid
A prismatoid is a polyhedron in which all vertices lie on exactly two parallel base planes. A base plane might contain only one or two vertices, but if it contains three or more, then that polygonal face is a base of the prismatoid. The height (or altitude) of a prismatoid is the perpendicular distance between the base planes.
Any prismatoid satisfies the necessary requirements for its volume to be given by the prismoidal volume formula: if hdenotes the height of the prismatoid, L and U denote the areas of the bases (where one of these areas might equal zero), and M is the area of a plane section parallel to a base and
midway between the base planes, then the volume,V, is given by Eq.


(The formula is not especially useful in practice because seldom is the area of the midsection known or easily determined.) Some solids other than prismatoids also have volumes given by this formula.


A prismoid is a prismatoid that satisfies some additional requirement. However, there is no agreement on precisely what that requirement is, so any use of the word should be accompanied by the user's choice from the many definitions appearing in dictionaries.


Prism
A prism is a prismatoid that has two congruent bases and whose other faces (the lateral faces) are all parallelograms. If the lateral faces are rectangles, then the prism is right; otherwise it is oblique. If B represents the area of either base and h represents the height, then the volume, V, of the prism is given by Eq.


A prism can be classified according to the type of polygonal region forming the bases: a triangular prism, a square prism, a hexagonal prism, and so forth. See also: Prism


Parallelepiped
A parallelepiped is a prism in which the bases are parallelograms. If the bases and all lateral faces are rectangles, then the prism is a rectangular parallelepiped or, more informally, a rectangular box. If l and w denote the length and width of a base, and h is the height of a rectangular parallelepiped, then the volume, V, is given by Eq. 


Cube
A cube is a rectangular parallelepiped in which all six faces are squares. If s denotes the length of an edge of the cube, then the volume, V, is given by Eq.


Pyramid
A pyramid is a prismatoid in which one base plane contains a single vertex: the vertex of the pyramid. The polygonal region defined by the vertices on the other base plane is the base of the pyramid. The base might contain any number of sides greater than two, but the other faces (the lateral faces) will all be triangular regions. A pyramid may be classified according to the shape of the base; thus a tetrahedron may also be called a triangular pyramid. A regular pyramid has a base that is a regular polygon, and its lateral faces are all congruent isosceles triangles. If Bdenotes the area of the base, and h is the height of the pyramid, then the volume, V, is given by Eq.


Frustrum
A frustrum (alternative name, frustum) of a pyramid is the solid remaining after a pyramid is sliced by a plane parallel to the base and the part containing the vertex is discarded. (The discarded part is another pyramid, similar to the original.) The bases of the frustrum are the base of the original pyramid and the new face created by the slicing. If Band b denote the areas of the bases, and h is the height of the frustrum, then the volume, V, is given by Eq.


Regular polyhedra
A polyhedron is regular (or platonic) if all faces are congruent and all dihedral angles (the angles between adjacent faces) are equal. There are only five regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron (see illustration and table).


The five regular polyhedra: (a) Tetrahedron. (b) Cube. (c) Octahedron. (d) Dodecahedron. (e) Icosahedron. 


The center of each face of a cube is the vertex of a regular octahedron; the center of each face of a regular octahedron is the vertex of a cube. The cube and the regular octahedron, therefore, are said to be reciprocal polyhedra. The regular dodecahedron and the regular icosahedron also form a pair of reciprocal polyhedra. The reciprocal of a regular tetrahedron is another regular tetrahedron