Combanatorics: An octagon has 8 sides. How many diagonals?

Question

I am missing something here. What do I need to know about octagons in order to form a combanatoric equation?

Answer 1

An octagon has 8 vertexes. you can draw a diagonal from each vertex to all but the two directly next to it. This is because drawing a line to the two next to it would be an edge not a diagonal. So there are are 5 vertexes that you can draw a diagonal to. 7 minus the two neighbors.

So you have 8 * 5 =40 possible diagonals. But each diagonal is connected to two vertexes. So you are counting them all twice. Divide 40 by 2 to get the actual number of diagonals, 20.

You can use this idea to find a formula to find the diagonals for a shape with n sides.

number of diagonals = n(n-2)/2

Answer 2

To derive the equation you need to generalize for all polygons. Start small and work up. A triangle has no diagonals, a square has 2, a pentagon has 5, a hexagon has 9 , etc. Tabulating these and the differences,
3 0
4 2 2
5 5 3 1
6 9 3 1
The 2nd column of differences is constant, so you have a quadratic relationship,
N = ax^2 + bx + c
Using simultaneous equations,
9a + 3b + c = 0
16a + 4b + c = 2
25a + 5b + c = 5
9a + b = 3
7a + b = 2
2a = 1
a = 1/2
7/2 + b = 4/2
b = -3/2
9/2 - 9/2 + c = 0
c = 0
so
N = (1/2)x^2 - (3/2)x
N = x(x - 3)/2
N(8) = 8*5/2 = 20
so all you need to know in the number of vertices.

Answer 3

20

Answer 4

an octagon has 8 sides means 4 diagonals cause diagonals
are opposite each other which means that a shape with an
odd number of sides cannot have diagonals

Answer 5

uh..octagons have 8 sides and 4 diagonals . why do you need an equation for that?

Answer 6

40/2=20

Answer 7

20 diagonals..