A convex polyhedron is called regular if its faces are regular polygons, each with the same number of sides, and for every vertex, the same number of edges converge.
There are five regular polyhedra, as follows:
Polyhedron | Planning | Elements |
Tetrahedron | 4 triangular faces 4 vertices 6 edges | |
Hexahedron | 6 square faces 8 vertices 12 edges | |
Octahedron | 8 triangular faces 6 vertices 12 edges | |
Dodecahedron | 12 pentagonal faces 20 vertices 30 edges | |
Icosahedron | 20 triangular faces 12 vertices 30 edges |
Euler's Relationship
In every convex polyhedron the following relation is valid:
V - A + F = 2
on what V is the number of vertices, THE is the number of edges and F, the number of faces. Take a look at the examples:
V = 8 A = 12 F = 6 8 - 12 + 6 = 2 | V = 12 A = 18 F = 8 12 - 18 + 8 = 2 |
Platonic polyhedra
A polyhedron is said to be platonic if and only if:
a) is convex;
b) at every vertex the same number of edges concur;
c) each face has the same number of edges;
d) the Euler relationship is valid.
Thus, in the figures above, the first polyhedron is platonic and the second non-platonic.
Next: Prisms