The 5 Regular Polyhedra

These 5 geometric figures are also known as the 5 Platonic Solids and are the only convex regular polyhedra that can exist.

A regular polyhedron is defined as a solid three-dimensional object having faces where
  • each face is a regular polygon. (A regular polygon has equal sides and equal angles).
  • the same number of faces (or the same number of edges) meet at each vertex
  • all the dihedral angles (the angles between the planes) are equal



  3 faces and 3 edges meet at each vertex

Each face is an equilateral triangle.

  4 faces

  4 vertices

  6 edges




  3 faces and 3 edges meet at each vertex

Each face is a square.

  6 faces

  8 vertices

  12 edges




  4 faces and 4 edges meet at each vertex

Each face is an equilateral triangle.

  8 faces

  6 vertices

  12 edges




  3 faces and 3 edges meet at each vertex

Each face is a pentagon.

  12 faces

  20 vertices

  30 edges




  5 faces and 5 edges meet at each vertex

Each face is an equilateral triangle.

  20 faces

  12 vertices

  30 edges


The Swiss mathematician Leonhard Euler (1707-1783) discovered the formula   V -E +F = 2   which states that the vertices minus the edges plus the faces of a convex polyhedron will always equal two.

If we were to inscribe a sphere within any of the 5 Platonic solids, it would be tangent to the center of each face.
If we were to circumscribe a sphere outside any of the 5 Platonic solids, it would pass through all of the vertices.


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