# Polyhedrons

# Polyhedrons

A polyhedron is a closed, three-dimensional solid bounded entirely by at least four polygons, no two of which are in the same plane. Polygons are flat, two-dimensional figures (planes) bounded by straight sides. A square and a triangle are two examples of polygons.

The number of sides of each polygon is the major feature distinguishing polyhedrons from one another. Some common polygons are the triangle (with three sides), the quadrilateral (with four sides), the pentagon (with five sides), the hexagon (with six sides), the heptagon (with seven sides), and the octagon (with eight sides).

A regular polygon, like the square, is one that contains equal interior angles and equal side lengths. A polygon is considered irregular if its interior angles are not equal or if the lengths of its sides are not equal.

Each of the polygons of a polyhedron is called a face. A straight side that intersects two faces is called an edge. A point where three or more edges meet is called a vertex. The illustration below indicates these features for a cube, which is a well-known polyhedron comprised of six square faces.

The relationship between the number of vertices (*v* ), faces (*f* ), and edges (*e* ) is given by the equation *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 polyhedron is called the Euler characteristic of the polyhedron's surface, named after the Swiss mathematician Leonhard Euler (1707–1783). Using the Euler characteristic and knowing two of the three variables, one can calculate the third variable.

## Platonic and Archimedean Solids

There are many groupings of polyhedrons classified by certain characteristics—too many to discuss here. One common group is known as the Platonic solids, so-called because its five members appeared in the writings of Greek philosopher Plato. The Platonic solids are within the larger grouping known as regular polyhedrons, in which the polygons of each are regular and congruent (that is, all polygons are identical in size and shape and all edges are identical in length), and are characterized by the same number of polygons meeting at each vertex.

The illustration below depicts the five Platonic solids (from left to right): tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

The tetrahedron consists of four triangular faces, and is represented as {3, 3}, in which the first 3 indicates that each face consists of three sides and the second 3 indicates that three faces meet at each vertex. The cube, sometimes called a hexahedron, has six square faces, and is represented as {4, 3}. The octahedron contains eight equilateral triangles, and is constructed by placing two identical square-based pyramids base to base. The octahedron is represented as {3, 4}. The dodecahedron consists of five sides to each face, and three pentagons meeting at each of the polyhedron' twenty vertices. It is represented by {5, 3}. The icosahedron is made by placing five equilateral triangles around each vertex. It contains congruent equilateral triangles for its twenty faces and twelve vertices, and is described as {3, 5}.

**Archimedean Solids.** Another common group of polyhedrons is the Archimedean solids, in which two or more different types of polygons appear. Each face is a regular polygon, and around every vertex the same polygons appear in the same sequence. For example, a truncated dodecahedron is made of the pentagon-pentagon-triangle sequence.

## Nets

A polyhedron can be "opened up" along some of its edges until its surface is spread out like a rug. The resulting map, similar to a dressmaker's pattern, is called a net. A net contains all faces of a polyhedron, some of them separated by angular gaps. Because a net is a flat pattern that can then be folded along the edges and taped together to regenerate the polyhedron of origin, a net therefore enables the easy construction of basic polyhedrons out of paper. The construction of polyhedron models can help make concepts in geometry easier to learn.

see also Nets.

*William Arthur Atkins with*

*Philip Edward Koth*

## Bibliography

Henderson, Kenneth B. *Modern Geometry: Its Structure and Function.* St. Louis: Webster Division McGraw-Hill Book Company, 1962.

# Polyhedron

# Polyhedron

A polyhedron is a three-dimensional closed surface or solid, bounded by plane figures called polygons.

The word polyhedron comes from the Greek prefix *poly-,* which means “many,” and the root word *hedron* which refers to “surface.” A polyhedron is a solid whose boundaries consist of planes. Many common objects are in the shape of polyhedrons. The cube is seen in everything from dice to clock radios; CD cases and sticks of butter are in the shape of polyhedrons called parallelpipeds. The pyramids are a type of polyhedron, as are geodesic domes. Most shapes formed in nature are irregular. In an interesting exception, however, crystals grow in mathematically perfect—and frequently complex— polyhedrons.

The bounding polygons of a polyhedron are called the faces. The line segments along which the faces meet are called the edges. The points at which the ends of edges intersect (think of the corner of a cereal box) are the vertices. Vertices are connected through the body of the polyhedron by an imaginary line called a diagonal.

A polyhedron is classified as convex if a diagonal contains only points inside of the polyhedron. Convex polyhedrons, also known as Euler polyhedrons, can be defined by the equation *E* = *v* + *f* –*e* = 2, where v is the number of vertices, *f* is the number of faces, and e is the number of edges. The intersection of a plane and a polyhedron is called the cross section of the polyhedron. The cross sections of a convex polyhedron are all convex polygons.

## Types of polyhedrons

Polyhedrons are classified and named according to the number and type of faces. A polyhedron with four sides is a tetrahedron, but is also called a pyramid. The six-sided cube is also called a hexahedron. A polyhedron with six rectangles as sides also has many names—a rectangular parallelepided, rectangular prism, or box.

A polyhedron whose faces are all regular polygons congruent to each other, whose polyhedral angels are all equal, and which has the same number of faces meet at each vertex is called a regular polyhedron. Only five regular polyhedrons exist: the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces—think of two pyramids placed bottom to bottom), the dodecahedron (12 pentagonal faces), and the icosahedron (20 triangular faces).

Other common polyhedrons are best described as the same as one of previously named that has part of it cut off, or truncated, by a plane. Imagine cutting off the corners of a cube to obtain a polyhedron formed of triangles and squares, for example.

Kristin Lewotsky

# Polyhedron

# Polyhedron

A polyhedron is a three-dimensional closed surface or solid, bounded by **plane** figures called **polygons** .

The word polyhedron comes from the Greek prefix *poly-* , which means "many," and the root word *hedron* which refers to "surface." A polyhedron is a solid whose boundaries consist of planes. Many common objects in the world around us are in the shape of polyhedrons. The cube is seen in everything from dice to clock-radios; CD cases, and sticks of butter, are in the shape of polyhedrons called parallelpipeds. The pyramids are a type of polyhedron, as are **geodesic** domes. Most shapes formed in nature are irregular. In an interesting exception, however, crystals grow in mathematically perfect, and frequently complex, polyhedrons.

The bounding polygons of a polyhedron are called the faces. The line segments along which the faces meet are called the edges. The points at which the ends of edges intersect (think of the corner of a cereal box) are the vertices. Vertices are connected through the body of the polyhedron by an imaginary line called a diagonal.

A polyhedron is classified as convex if a diagonal contains only points inside of the polyhedron. Convex polyhedrons are also known as Euler polyhedrons, and can be defined by the equation *E* = *v* + *f*- *e* = 2, where *v* is the number of vertices, *f* is the number of faces, and *e* is the number of edges. The intersection of a plane and a polyhedron is called the **cross section** of the polyhedron. The cross-sections of a convex polyhedron are all convex polygons.

## Types of polyhedrons

Polyhedrons are classified and named according to the number and type of faces. A polyhedron with four sides is a **tetrahedron** , but is also called a **pyramid** . The six-sided cube is also called a hexahedron. A polyhedron with six rectangles as sides also has many names—a rectangular parallelepided, rectangular **prism** , or box.

A polyhedron whose faces are all regular polygons congruent to each other, whose polyhedral angels are all equal, and which has the same number of faces meet at each vertex is called a regular polyhedron. Only five regular polyhedrons exist: the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces—think of two pyramids placed bottom to bottom), the dodecahedron (12 pentagonal faces), and the icosahedron (20 triangular faces).

Other common polyhedrons are best described as the same as one of previously named that has part of it cut off, or truncated, by a plane. Imagine cutting off the corners of a cube to obtain a polyhedron formed of triangles and squares, for example.

Kristin Lewotsky

# polyhedron

pol·y·he·dron / ˌpäliˈhēdrən/ • n. (pl. -he·drons or -he·dra / -ˈhēdrə/ ) Geom. a solid figure with many plane faces, typically more than six.DERIVATIVES: pol·y·he·dral / -ˈhēdrəl/ adj.pol·y·he·dric / -ˈhēdrik/ adj.

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