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

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Presentation on theme: "E + 2 = F + V The Euler’s Formula for polyhedra. E = number of edges F = number of faces V = number of vertices."— Presentation transcript:

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

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 base 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 base 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 base 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 base 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 base E vs. F + V 16 vs

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

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

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

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

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

25 8 edges of the octagonal base correspond to8 lateral faces 2 lateral edgescorrespond to vertices at the base face at the bottom 1 vertex at the top 16 + = E vs. F + V 16 vs vs vs 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 base face at the bottom 1 vertex at the top 16 + = E vs. F + V 16 vs vs vs 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


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