## Diagonals in Octagons

How many more diagonals are in an octagon than in a hexagon?
Source: NCCTM Mathematics Teacher

SOLUTION

We draw a diagonal by connecting two non-adjacent vertices. For example, $1,3$ is a diagonal but $1,2$ and $1,8$ are not. How many ways can we choose two vertices from a set of eight vertices?
$\binom{8}{2}=28$
1. $1,2$ N    8. $2,3$ N    14. $3,4$ N    19. $4,5$ N    23. $5,6$ N    26. $6,7$ N    28. $7,8$ N
2. $1,3$ Y    9. $2,4$ Y     15. $3,5$ Y    20. $4,6$ Y    24. $5,7$ Y     27. $6,8$ Y
3. $1,4$ Y    10. $2,5$ Y   16. $3,6$ Y    21. $4,7$ Y     25. $5,8$ Y
4. $1,5$ Y    11. $2,6$ Y    17. $3,7$ Y    22. $4,8$ Y
5. $1,6$ Y    12. $2,7$ Y    18. $3,8$ Y
6. $1,7$ Y    13. $2,8$ Y
7. $1,8$ N
Number of diagonals in an octagon
$28-8=20$
How many ways can we choose two vertices from a set of six vertices of a hexagon?
$\binom{6}{2}=15$
1. $1,2$ N    6. $2,3$ N    10. $3,4$ N    13. $4,5$ N    15. $5,6$ N
2. $1,3$ Y    7. $2,4$ Y     11. $3,5$ Y     14. $4,6$ Y
3. $1,4$ Y    8. $2,5$ Y    12. $3,6$ Y
4. $1,5$ Y    9. $2,6$ Y
5. $1,6$ N
Number of diagonals in a hexagon
$15-6=9$
How many more diagonals are in an octagon than in a hexagon?
$20-9=11$

Answer: $11$