Polyhedron

A polyhedron is a three dimensional region of space bounded by polygons.

Components of a Polyhedron

Faces

The faces of a polyhedron are each of the two dimensional polygons that border the polyhedron.

Edges

The edges of a polyhedron are the sides of the faces of the polyhedron. Two faces have an edge in common.

Vertices

The vertices of a polyhedron are the vertices of each of the faces of the polyhedron. Three faces coincide with the same vertex.

Dihedral Angles

The dihedral angles are formed between two faces of all neighboring polygons.

Polyhedral Angles

Polyhedral angles are formed by three or more faces of the polyhedron and have a common vertex.

Diagonals

The diagonals of a polyhedron are the line segments joining two vertices not belonging to the same face.


Euler's formula

It is verified that in all convex polyhedra:

No. of faces + No. of vertices = No. of de edges + 2.

Types of Polyhedra

Convex Polyhedron

Convex Polyhedron


In a convex polyhedron, a straight line can only penetrate the surface in two points.

Concave Polyhedron

Concave Polyhedron


In a concave polyhedron, a straight line can pentrate the surface in more than two points.

Regular Polyhedra

A regular polyhedron is composed of angles and faces (regular polygons) that are all equal.

Platonic Solids

The platonic solids are convex regular polyhedra. There are exactly five types of platonic solids:

Tetrahedron.

Hexahedron or cube.

Octahedron.

Dodecahedron.

Icosahedron.

Irregular Polyhedra

An irregular polyhedron is defined by polygons that are composed of elements that are not all equal.

Types of Polyhedra by the Number of Faces

Tetrahedron

Polyhedron of 4 faces.

Pentahedron

Polyhedron of 5 faces.

Hexahedron

Polyhedron of 6 faces.

Heptahedron

Polyhedron of 7 faces.

Octahedron

Polyhedron of 8 faces.

Nonahedron

Polyhedron of 9 faces.

Decahedron

Polyhedron of 10 faces.

Undecahedron

Polyhedron of 11 faces.

Dodecahedron

Polyhedron of 12 faces.

Icosahedron

Polyhedron of 20 faces.