# Polyhedron

In Geometry, a polyhedron is a closed space figure whose faces are polygons. The word polyhedron has Greek origins, meaning many faces. The following are a few examples of polyhedra.

## Characteristics of a polyhedron

The polygons that form a polyhedron are called faces. The line segments created by two intersecting faces are called edges. The vertices are points where three or more edges meet.

The hexagonal prism above is a polyhedron that has 6 lateral faces that are parallelograms, and 2 faces on the top and bottom, called bases, that are hexagons.

## Euler's Theorem

Euler's Theorem shows a relationship between the number of faces, vertices, and edges of a polyhedron. It states that the sum of the faces and vertices minus the number of edges always equals two:

F + V - E = 2

where F is the number of faces, V is the number of vertices, and E is the number of edges of a polyhedron.

Example:

For the hexagonal prism shown above, F = 8 (six lateral faces + two bases), V = 12, and E = 18:

8 + 12 - 18 = 2

## Classifications of polyhedra

Polyhedra can be classified in many ways. For example, they can be classified as regular and irregular polyhedra. A regular polyhedron is a polyhedron whose faces are all congruent regular polygons; any polyhedron that does not meet these conditions is considered irregular.

Polyhedra can also be classified as convex and concave. A concave polyhedron has at least one face that is a concave polygon. A polyhedron that is not concave, is convex. Polyhedra can also be classified based on the number of faces it has. For example, a tetrahedron has 4 faces, a pentahedron has 5 faces, and a hexahedron has 6 faces.

The following is a list of terms often used to describe polyhedra based on their characteristics.

### Prisms

Prisms are polyhedra that have two congruent faces, called bases, lying in parallel planes. A prism is typically named by the shape of its polygonal bases. The lateral faces (the sides that are not bases) are parallelograms, rectangles, or squares.

Regular prism | Irregular prism |
---|---|

The bases for the regular hexagonal prism above have bases that are regular hexagons. | The bases for the hexagonal prism above are irregular hexagon. |

### Pyramids

Pyramids are polyhedra that has a polygon as its base and triangles as all its other faces. A pyramid is also typically named by the shape of its polygonal base.

Regular pyramid | Irregular pyramid |
---|---|

The base of the square pyramid above has a base that is a square (a regular polygon). | The base for the trapezoidal pyramid above is a trapezoid with unequal sides (so it is an irregular polygon). |

### Regular polyhedra

A regular polyhedron is a polyhedron whose faces are all congruent, regular polygons. A regular polyhedron is named based on its number of faces. There are only five polyhedra that are regular polyhedra; these are referred to as Platonic solids.

**The five Platonic solids**

In the diagram above, each regular polyhedra is named based on its number of faces. The net below each sketch shows a 2D picture of all of the faces of the polyhedron.

Most regular prisms are generally not considered regular polyhedra. A cube is the only regular prism that can also be classified as a regular polyhedron.

Likewise, a regular tetrahedron is the only regular pyramid that is also a regular polyhedron.