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

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

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

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A net for an orthogonal pyramid

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1 edge of the octagonal base corresponds to1 lateral face

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2 edges of the octagonal base correspond to lateral faces 2

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

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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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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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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