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

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About mvtrinh

Retired high school math teacher.
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