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Pictures of Platonic Solids

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Why only five Platonic Solids?

dodecahedron Paper model dodecahedron on pedestal

Dodecahedron
Number of faces: 12
Number of edges: 30
Number of vertices: 20

tetrahedron paper model Paper model tetrahedron on pedestal

Tetrahedron
Number of faces: 4
Number of edges: 6
Number of vertices: 4

cube paper model Paper model cube on pedestal

Cube
Number of faces: 6
Number of edges: 12
Number of vertices: 8




octahedron paper model Paper model octahedron on pedestal

Octahedron
Number of faces: 8
Number of edges: 12
Number of vertices: 6

icosahedron Paper model icosahedron on pedestal

Icosahedron
Number of faces: 20
Number of edges: 30
Number of vertices: 12




Platonic Solids
Platonic Solids dice numbers
Platonic Solids dice dots
Platonic Solids Platonic Solids dice numbers Platonic Solids dice
Platonic Solids (.PDF) Platonic Solids numbers (.PDF)
Platonic Solids dots (.PDF)


Platonic Solids
There are five so named because they were known at the time of Plato circa (427-347 BC).  These polyhedra are also called regular polyhedra because they are made up of faces that are all the same regular polygon.


Why only five Platonic Solids

Number Faces of Platonic Solids at a vertex A Platonic solid is a polyhedron all of whose faces are congruent regular convex polygons*, and where the same number of faces meet at every vertex.

The Greeks recognized that there are only five platonic solids. But why is this so?
The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees.

Tetrahedron:
Three triangels at a vertex: 3*60 = 180 degrees

Octahedron:
Four triangles at a vertex: 4*60 = 240 degrees

Icosahedron:
Five triangles at a vertex: 5*60 = 300 degrees

Cube:
Three squares at a vertex: 3*90 = 270 degrees

Dodecahedron:
Three pentagons at a vertex: 3*108 = 324 degrees

Note:
  Six triangles: 6*60 = 360 degrees
  Four squares: 4*90 = 360 degrees
  Four pentagons: 4*108 = 432 degrees
  Three hexagons: 3*120 = 360 degrees
So there are only five Platonic Solids!

 

 

*) Regular means that the sides of the polygon are all the same length.
Congruent
means that the polygons are all the same size and shape.

External Links:
Mathforum dr.math faq Regular Polyhedra
Mathworld: Platonic Solid
University of Utah: Platonic Solids
Mathsisfun: Platonic Solids
Dartmouth College: Platonic Solids Unit6
Science U: Solids
Wikipedia: Platonic Solids




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