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E + 2 = F + V
The Euler’s Formula for polyhedra.
E = number of edgesF = number of facesV = 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 to 1 lateral face
2 edges of the octagonal base
correspond to lateral faces2
22 edges of the octagonal base
correspond to lateral faces3 3
2 edges of the octagonal base
correspond to lateral faces4 4
2 edges of the octagonal base
correspond to lateral faces23 34 45 56 67 78 8
8 edges of the octagonal base
correspond to 8 lateral faces
1 lateral edge corresponds to 1 vertex at the base
1st edge
1st vertex
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base2
1st edge
2nd edge
2nd vertex
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233
1st edge
2nd edge
3rd edge
3rd vertex
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base244
1st edge
2nd edge
3rd edge
4th edge
4th vertex
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 55
5th vertex
5th edge
1st edge
2nd edge
3rd edge
4th edge
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 66
6th vertex
6th edge
5th edge
1st edge
2nd edge
3rd edge
4th edge
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 77
7th vertex
7th edge
6th edge
5th edge
1st edge
2nd edge
3rd edge
4th edge
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
8th vertex
8th edge
7th edge
6th edge
5th edge
1st edge
2nd edge
3rd edge
4th edge
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
E vs. F + V
16 vs. 8 + 8
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
E + 2 = F + V
But 16 + 2 ≠ 8 + 8
How can we account for the
2?
E vs. F + V
16 vs. 8 + 8
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
1 face at the bottom
Bottom face
E vs. F + V
16 vs. 8 + 8
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
E vs. F + V
16 vs. 9 + 8
1 face at the bottom
Bottom face
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
1 face at the bottom
Bottom face
1 vertex at the top
E vs. F + V
16 vs. 9 + 8
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
1 face at the bottom
1 vertex at the top
E vs. F + V
16 vs. 8 + 816 vs. 9 + 816 vs. 9 + 9
Bottom face
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
1 face at the bottom
1 vertex at the top
16 + = 9 + 92
E vs. F + V
16 vs. 8 + 816 vs. 9 + 816 vs. 9 + 9
16 is not equal to 18! What do you need to add to
make both sides equal.
8 edges of the octagonal base
correspond to 8 lateral faces
2 lateral edges correspond to vertices at the base233 44 55 66 77 88
1 face at the bottom
1 vertex at the top
16 + = 9 + 92
E vs. F + V
16 vs. 8 + 816 vs. 9 + 816 vs. 9 + 9
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
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