Polyhedra

The 2-dimensional shapes of a polygon can be applied in a 3-dimensional figure. Such characteristics define polyhedra. Polyhedron is a very general terms and can include some very complex shapes.

**Polyhedron **(plural, polyhedra)**: **A three-dimensional figure made up with polygon faces

**Face: **A polygon in a polyhedron.

**Lateral Face: **A face that is not the base.

**Edge: **The line segment where two lateral faces intersect.

**Lateral Edge: **The line segment where two lateral faces intersect.

**Vertex **(plural, vertices)**: **The point where two edges intersect.

**Regular Polyhedron: **A polyhedron where all the faces are congruent regular polygons.

A** polyhedron** has these properties:

- 3-dimensional
- Made of only flat polygons, called the
**faces**of the polyhedron - Polygon faces join together along segments called
**edge** - Each edge joins exactly two faces
- Edges meet in points called
**vertices;**each edge joins exactly two vertices - There are no gaps between edges or vertices
- Can be convex or concave

Two common types of polyhedra include prisms and pyramids. Prisms and pyramids are named by their bases.

*Prism:*A polyhedron with two parallel, congruent bases. The other faces, also called lateral faces, are formed by connecting the corresponding vertices of the bases.- Left: triangular prism Right: octagonal prism

*Pyramid:*A polyhedron with one base and triangular sides meeting at a common vertex.- Left: hexagonal pyramid Right: square pyramid

Polyhedra Cont.

This formula can be used to find the number of vertices (V), faces (F), or edges (E) on a polyhedron:

** F + V = E + 2**

If a figure does not satisfy Euler’s formula, the figure is not a polyhedron.

A** regular polyhedron** has the following characteristics:

- All faces are congruent regular polygons
- Satisfies Euler’s formula for the number of vertices, faces, and edges
- The figure has no gaps or holes
- The figure is convex (has no indentations)

Named after the Greek philosopher Plato, the five regular polyhedra are:

- regular tetrahedron: 4-faced polyhedron where all the faces are equilateral triangles
- cube: 6-faced polyhedron where all the faces are squares
- regular octahedron: 8-faced polyhedron where all the faces are equilateral triangles
- regular dodecahedron: 12-faced polyhedron where all the faces are regular pentagons
- regular icosahedron: 20-faced polyhedron where all the faces are equilateral triangles

A polyhedron is semi-regular if all of its faces are regular polygons and satisfies Euler’s formula.

- Semi-regular polyhedra often have two different kinds of faces, both of which are regular polygons.
- Prisms with a regular polygon base are one example of semi-regular polyhedron.