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Regular polyhedra


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