Geometry

Polyhedra

Big Picture

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.

Key Terms

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.

Classifying Polyhedra

A polyhedron has these properties:

  1. 3-dimensional
  2. Made of only flat polygons, called the faces of the polyhedron
  3. Polygon faces join together along segments called edge
  4. Each edge joins exactly two faces
  5. Edges meet in points called vertices; each edge joins exactly two vertices
  6. There are no gaps between edges or vertices
  7. Can be convex or concave
Classifying Polyhedra

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
Prism
  • Pyramid: A polyhedron with one base and triangular sides meeting at a common vertex.
  • Left: hexagonal pyramid Right: square pyramid
Pyramid
Image Credit: Blue figures on this page copyright rtguest, 2014, modified by CK-12 Foundation. Used under license from Shutterstock.com.

Geometry

Polyhedra Cont.

Euler’s Formula for Polyhedra

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.

Regular Polyhedra

A regular polyhedron has the following characteristics:

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

Platonic Solids

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

  1. regular tetrahedron: 4-faced polyhedron where all the faces are equilateral triangles
  2. cube: 6-faced polyhedron where all the faces are squares
  3. regular octahedron: 8-faced polyhedron where all the faces are equilateral triangles
  4. regular dodecahedron: 12-faced polyhedron where all the faces are regular pentagons
  5. regular icosahedron: 20-faced polyhedron where all the faces are equilateral triangles
Platonic Solids

Semi-Regular Polyhedra

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.

Notes