1. E + 2 = F + V The Euler’s Formula for polyhedra. E = number of edges F = number of faces V = number of vertices

2. E + 2 = F + V We’ll illustrate why the Euler’s Formula works for an orthogonal pyramid (a pyramid with an 8-sided base).

3. E + 2 = F + V It’s easier to see using a 2-dimensional representation of an orthogonal pyramid.

4. Can you draw a net for an orthogonal pyramid? It’s easier to see using a 2-dimensional representation of an orthogonal pyramid.

5. A net for an orthogonal pyramid

6. 1 edge of the octagonal base corresponds to1 lateral face

7. 2 edges of the octagonal base correspond to lateral faces 2

8. 2 2 edges of the octagonal base correspond to lateral faces 3 3

9. 2 edges of the octagonal base correspond to lateral faces 4 4

10. 2 edges of the octagonal base correspond to lateral faces 2 3 3 4 4 5 5 6 6 7 7 8 8

11. 8 edges of the octagonal base correspond to8 lateral faces 1 lateral edge corresponds to1 vertex at the base 1 st edge 1 st vertex

12. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2 1 st edge 2 nd edge 2 nd vertex

13. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base233 1 st edge 2 nd edge 3 rd edge 3 rd vertex

14. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base244 1 st edge 2 nd edge 3 rd edge 4 th edge 4 th vertex

15. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667755 5 th vertex 5 th edge 1 st edge 2 nd edge 3 rd edge 4 th edge

16. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667766 6 th vertex 6 th edge 5 th edge 1 st edge 2 nd edge 3 rd edge 4 th edge

17. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667777 7 th vertex 7 th edge 6 th edge 5 th edge 1 st edge 2 nd edge 3 rd edge 4 th edge

18. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 8 th vertex 8 th edge 7 th edge 6 th edge 5 th edge 1 st edge 2 nd edge 3 rd edge 4 th edge

19. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 E vs. F + V 16 vs. 8 + 8

20. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 E + 2 = F + V But 16 + 2 ≠ 8 + 8 How can we account for the 2? E vs. F + V 16 vs. 8 + 8

21. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 1 face at the bottom Bottom face E vs. F + V 16 vs. 8 + 8

22. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 E vs. F + V 16 vs. 9 + 8 1 face at the bottom Bottom face

23. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 1 face at the bottom Bottom face 1 vertex at the top E vs. F + V 16 vs. 9 + 8

24. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 1 face at the bottom 1 vertex at the top E vs. F + V 16 vs. 8 + 8 16 vs. 9 + 816 vs. 9 + 9 Bottom face

25. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 1 face at the bottom 1 vertex at the top 16 + = 9 + 9 2 E vs. F + V 16 vs. 8 + 8 16 vs. 9 + 816 vs. 9 + 9 16 is not equal to 18! What do you need to add to make both sides equal.

26. 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base2334455667788 1 face at the bottom 1 vertex at the top 16 + = 9 + 9 2 E vs. F + V 16 vs. 8 + 8 16 vs. 9 + 816 vs. 9 + 9 E + 2 = F + V

27. We have derived the Euler formula using an orthogonal pyramid (i.e. a pyramid with an 8-sided base). E + 2 = F + V

28. Now, use this idea to prove that the Euler’s formula works for all pyramids with a polygonal base. We have derived the Euler formula using an orthogonal pyramid (i.e. a pyramid with an 8-sided base). E + 2 = F + V