Expert: Scott A Wilson - 10/13/2009

Question
How does the number of diagonals from each vertex compare to the number of sides?

Answer
In a triagle, there are none.
In a square, there are two.
In a pentagram, there are five.
In a hexagon, there are nine.
We can make the following difference table.

x y
3 0
4 2 2
5 5 3 1/2
6 9 4 1/2

For the first column of differences, take d(i+1) = (y(i+1)-y(i))/(x(i+1)-x(i)).
For the second column, take e(i+1) = (d(i+1) - d(i))/(x(i+1)-x(i-1)).

The equation is then ((1/2)(x-4)+2)(x-3)+0.
Since (1/2)(x-4) is x/2 - 2, adding 2 makes it x/2.

This is multiplied by (x-3), so the result is N(x) = x(x-3)/2
is the number of sides on a n-gon.

3: 3(0)/2 = 0
4: 4(1)/2 = 2
5: 5(2)/2 = 5
6: 6(3)/2 = 9

Yeah, that checks.