Answer:
Explanation:
Let
V
be the number of vertices of a polyhedron,
F
the number of faces of that polyhedron, and
E
be the number of edges. The quantity
χ
=
V
−
E
+
F
is called the Euler characteristic (of a polyhedron). In the case of convex polyhedra,
χ
=
2
.
Consider, for example, a tetrahedron (which is the simplest solid). It has 4 faces,
1
2
(
4
)
(
3
)
=
6
edges, and
1
3
(
4
)
(
3
)
=
4
vertices. Thus we have
V
−
E
+
F
=
4
−
6
+
4
=
2
.
Euler's formula holds for all Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Since a cube and an octahedron are dual polyhedra (each is formed by connecting the centers of the faces of the other), their
V
and
F
values are equal to the
F
an
V
values of the other. (The same is true for the dodecahedron and icosahedron).