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

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

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

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

A net for an orthogonal pyramid

1 edge of the octagonal base corresponds to1 lateral face

2 edges of the octagonal base correspond to lateral faces 2

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

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

2 edges of the octagonal base correspond to lateral faces

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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