Combinations

[ssw513]

How can you use combinations to find the number of diagonals of a 16 sided polygon? I am not sure because I tested a my equation with a square and it was more than 2. Would the polygon be considered a circular function?

 

[galactus]

The number of diagonals in a polygon is given by \(\displaystyle \L\\\frac{n(n-3)}{2}\)

That's because each vertex can join n-3 lines and there are n of them.

But don't count them twice, so divide by 2.

 

[pka]

It is the combination of the number of vertices taken two at a time.
But you then need to subtract the number of edges.

 

[TchrWill]

How can you use combinations to find the number of diagonals of a 16 sided polygon? I am not sure because I tested a my equation with a square and it was more than 2. Would the polygon be considered a circular function?

The number of diagonals in the first series of polygons are

Number of sides.........n = 3....4....5....6....7....8
Number of diagonals.....N = 0....2....5....9...14..20
1st Difference.................2....3....4....5....6
2nd Difference....................1....1....1....1

We therefore, have a finite difference sequence with the 2nd differences constant at 1. This means that the general expression for the number of diagonals in any n-gon is of the form N = an^2 + bn + c.

Using the data, we can write
a(3^2) + b(3) + c = 0 or 9a + 3b + c = 0
a(4^2) + b(4) + c = 2 or 16a + 4b + c = 2
a(5^2) + b(5) + c = 5 or 25a + 5b + c = 5

Solving this set of equations leads us to a = 1/2, b = -3/2, and c = 0 resulting in N = n^2/2 - 3n/2 = n(n - 3)/2.

 

[ssw513]

Thanks

we thought about the edges after the post. thanks we got 104