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Platonic Solids

The platonic solids (or regular polyhedra) are convex with faces composed of congruent , convex regular polygons .  The mathematician Euclid proved that there are exactly five such solids.  They are the tetrahedron, cube, octahedron, dodecahedron and icosahedron.

The tetrahedron has 6 faces.  Each is an equilateral triangle .  It also has 6 edges and 6 vertices.  At each vertex three edges meet.

      Surface Area = 3 e 2

      Volume = 2 12 e 3

The cube has 6 faces.  Each is a square .  It also has 12 edges and 8 vertices.  At each vertex three edges meet.

      Surface Area = 6 e 2
      Volume = e 3

The octahedron has 8 faces.  Each is an equilateral triangle.  It also has 12 edges and 6 vertices.  At each vertex four edges meet.

      Surface Area = 2 3 e 2

      Volume = 2 12 e 3

The dodecahedron has 12 faces.  Each is a regular pentagon.  It also has 30 edges and 20 vertices.  At each vertex three edges meet.

      Surface Area = 3 25 + 10 5 e 2

      Volume = 15 + 7 5 4 e 3

The icosahedron has 20 faces.  Each is an equilateral triangle.  It also has 30 edges and 12 vertices.  At each vertex five edges meet.

      Surface Area = 5 3 e 2

      Volume = 5 12 ( 3 + 5 ) e 3