2. How many vertices, edges, and faces are contained in each of the polyhedra? vertices of each polygon polygons meeting at a vertex faces of the polyhedron edges of the polyhedron vertices of the polyhedron Tetrahedron 33464 Octahedron 348126 Icosahedron 35203012 Hexahedron 431268 Dodecahedron 53123020

3. What is the relationship between the number of vertices, faces and edges? Euler’s formula

4. How would you measure the angles of these polyhedra? What are the angle measures of each? What is the total measure of each? One way is using the formula (n-2)180 where n = number of sides, then multiply by the number of faces. Students may come up with other ways to the answer such as the number of polygon vertices around a point times the number of polyhedron vertices Equilateral Triangle - 180 degrees (60 per angle) x number of faces Square - 360 degrees (90 per angle) x number of faces Pentagon - 540 degrees (108 per angle) x number of faces

5. Where did these shapes originate in nature? The Tetrahedron, Hexahedron and Octahedron come from crystals. The Dodecahedron and Icosahedron come from microscopic animals called Radiolarians. The Herpes virus is in the shape of the Icosahedron.

6. What did these platonic solids represent in mythology? Tetrahedron - Fire Hexahedron - Earth Octahedron - Air Dodecahedron - Universe Icosahedron - Water

7. Why are there only 5 platonic solids? The interior angles of the polygon must add up to at less than 360 degrees. The icosahedron is 5 x 60 degrees. One more would give 360 degrees which would be a flat circle. Any more pentagons or squares would not fit around a point.

8. Concerning architecture, which platonic solids would be most sturdy? Why? The tetrahedron has held up better in architecture. Most houses are based on the triangle. The triangle withstands much more pressure than the other shapes.

9. Vertices – Edges + Faces = 2

10. Four triangular faces, four vertices, and six edges

11. Six square faces, eight vertices, and twelve edges

12. Eight triangular faces, six vertices, and twelve edges.

13. Twelve pentagonal faces, twenty vertices, and thirty edges.

14. Twenty triangular faces, twelve vertices, and thirty edges