| 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. |
