3.

PRISM
Number of Sides in Base
Vertices
Faces
Edges
Triangular 3 6 5 9
Rectangular 4 8 6 12
Pentagonal 5 10 7 15
Hexagonal 6 12 8 18
Octagonal 8 16 10 24
PYRAMID
Number of Sides in Base
Vertices
Faces
Edges
Triangular 3 4 4 6
Rectangular 4 5 5 8
Pentagonal 5 6 6 10
Hexagonal 6 7 7 12
Octagonal 8 9 9 16

For prisms:

The number of vertices is the number of vertices in the base times two. This is because a prism is just the base drawn twice, with the vertices connected. The number of faces is the number of sides in the base plus two (one face for each edge plus the front and back faces). The number of edges is the number of vertices plus the number of faces, minus two. In mathematical language, these relationships would be written as follows:

  1. # of Vertices = # of vertices in the base x 2
  2. # of Faces = # of sides in the base + 2
  3. # of Edges = # of vertices + # of faces - 2

For pyramids:

The number of vertices is the number of sides in the base plus one. This is because a pyramid is just the base with each vertex connected to a point not on the base. The number of faces is the same as the number of vertices (one face for each of the sides in the base plus the base itself). Again, the number of edges is the number of vertices plus the number of faces, minus two. In 1751, a great mathematician by the name of Leonhard Euler (1707-1783) proved that this is true for all convex polyhedra . It is now known as "Euler's Formula", and is written as F + V = E + 2 (Faces + Vertices = Edges + 2). This can be rewritten as V + F - 2 = E, which is what we noticed in our tables above.

In mathematical language, these relationships would be written as follows:

  1. # of Vertices = # of sides in the base + 1
  2. # of Faces = # of vertices
  3. # of Edges = # of vertices + # of faces - 2