Polyhedron

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The faces of a polyhedron are made up of polygons. The faces of a regular polyhedron are equal in size and shape. This illustration depicts the only five regular polyhedra, with their names and number of sides.

Polyhedron, in geometry, a solid bounded by flat surfaces with each surface bounded by straight sides. In other words, a polyhedron is a solid bounded by polygons. Each of the flat surfaces is called a face. A straight side bounding a face is called an edge. A point at the end of an edge is called a vertex. Figure 1, a pyramid with a square base and four triangular sides, is an example of a polyhedron.

In a regular polyhedron all of the faces are regular polygons that are congruent (equal in size and shape). The only regular polyhedra are the five shown in figure 2. They are the tetrahedron, which has four triangular faces; the cube, which has six square faces; the octahedron, which has eight triangular faces; the dodecahedron, whose 12 faces are all regular pentagons; and the icosahedron, which has 20 triangular faces. These are sometimes referred to as the Platonic solids because they appear in the writing of the Greek philosopher Plato, representing fire, air, earth, water, and the universe as a whole.

A convex polyhedron is one in which a line segment connecting any two vertices of the polyhedron contains only points that are on a face or inside the polyhedron. For convex polyhedrons, the relationship between the number of vertices v, faces f and edges e is given by v + f - e = 2. For example, the cube has 8 vertices, 6 faces, and 12 edges, which gives 8 + 6 - 12 = 2. The value of v + f - e for a general polyhedron is called the Euler characteristic of the surface of the polyhedron, named after the Swiss mathematician Leonhard Euler. It can be calculated for general polyhedra using the methods of algebraic topology, a branch of mathematics.

Contributed By:
William James Ralph
yramid (geometry)
Article
Pyramid (geometry), solid figure formed by connecting every point on or interior to a plane polygon to a single point not in the plane (see Fig. 1 and Fig. 2). A pyramid is thus a special case of a cone Contents.asp?pg=2&ti=761552932> or of a polyhedron, a solid bounded by planes. The polygon (in Fig. 1, ABC; in Fig. 2, DEFGH) is the base of the pyramid, and the point V (or W) is the apex or vertex; the line segments, such as VA and VB, are the lateral edges of the pyramid, and the triangular sides, such as VAB and VBC, are the lateral faces. The altitude of a pyramid is the perpendicular distance from the vertex to the plane of the base.

The name of a pyramid depends upon the shape of its base. For example, a square pyramid has a square base, while a hexagonal pyramid has a six-sided base. A triangular pyramid, Fig. 1, is also called a tetrahedron; it is bounded by four triangles, any one of which may be considered the base.

A regular pyramid has a regular polygon as the base, with the vertex perpendicular to the base at its center; the slant height of a regular pyramid is the altitude (from the vertex) of any lateral face. A frustum of a pyramid is the solid between the base and a plane parallel to the base, as in Fig. 3. A truncated pyramid is the solid between the base and a plane cutting all lateral edges, as in Fig. 4.

The lateral area of a pyramid is the sum of the areas of the lateral faces; in particular, the lateral area of a regular pyramid is sp/2, in which s is the slant height and p is the perimeter of the base. The volume of any pyramid is hK/3, in which h is the altitude of the pyramid and K is the area of the base. The volume of a pyramid is thus one-third of the volume of a prism that has the same base and altitude.

Contributed By: James Singer, M.A., Ph.D.
Late Professor Emeritus of Mathematics, Brooklyn College of the City University of New York. Author of Elements of Numerical Analysis

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