# A certain polyhedron has 6 vertices and 9 edges. Determine the number of faces on this polyhedron.

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Let x = the number of faces on the polyhedron.

Now, according to Euler's polyhedron formula:

Number of vertices - number of edges+ number of faces =2

==> 6 - 9 +x = 2

==> x = 5

then the polyhedron has 5 faces.

Descates (in 1639) has given us the relation between the vertices , faces and edges of any polyhedron or any plane figure bounded by sraight edges, plane polygonal faces and vertices.It is popularly known as Euler's formula(Euler rediscovered it in 1751) or Euler's theorem and proved by Cahchy also in 1811 :

The number of vertices+ number of faces = number of edges +2. Or V+F = E+2. Substitute the given values , V=6 and E = 9 and we get: 6+F = 9+2 . So F = 9+2-6 = 5..