How many diagonals does a regular 20 sided figure have?

[atrain4life]

I need help with this problem...

How many diagonals does a regular polygon with 20 sides have?

thank you in advance

 

[skeeter]

# of sides...# of diagonals
3 ................... 0
4 ................... 2
5 ................... 5
6 ................... 9

see a pattern?

 

[galactus]

There's a formula:

\(\displaystyle \L\\\frac{n(n-3)}{2}\)

 

[TchrWill]

How many diagonals does a regular polygon with 20 sides have?

How many diagonals are there within any polygon? Is there a formula for determining the number of diagonals?

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.

 

[pka]

Here is a straightforward way.
Given n points no three of which are collinear, these points determine \(\displaystyle {n \choose 2} = \frac{{n(n - 1)}}{2}\) line segments. Of those n are the sides of the polygon therefore \(\displaystyle \frac{{n(n - 1)}}{2} - n = \frac{{n^2 - 3n}}{2}.\)

This is exactly the formula galactus has given you with reasons added.

 

[atrain4life]

Thank you for helping me find that formula! I knew I had to find a formula, but I'm not very good at that part.

Thanks again!