Exercise 10 (Chapter Review)

Your friend has challenged you to create a convex polyhedron containing 9 triangles and 6 pentagons.

1. Is it possible to build such a polyhedron using only these shapes? Explain.

2. You decide to also include one heptagon (seven-sided polygon). How many vertices does your new convex polyhedron contain?

3. Assuming you are successful in building your new 16-faced polyhedron, could every vertex be the joining of the same number of faces? Could each vertex join either 3 or 4 faces? If so, how many of each type of vertex would there be?