an octagon has 8 sides. how many diagonal does an octagon have

Answer 1

A triangle has 0 diagonals
A quadrilateral has 2 diagonals
A pentagon has 5 diagonals
A hexagon has 9 diagonals

Can you see a pattern ???

or, think of it this way

A hexagon has 6 vertices.
Joining any 2 of those 6 will produce either a side or a diagonal
The number of possible lines I can draw is C(6,2)
(choose any two points from the 6), which would be 15
But that would include the 6 original sides, so the rest or 9, must be diagonals.

Use the same argument for any n-gon

the number of diagonals for an n-gon would be
C(n,2) - n

(test this for my previous answers above)