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