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

pianoplaya16 said:
How many diagonals does a convex polygon with 20 sides have?
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What formulas did they give you? What have you tried so far?

Please be specific. Thank you.

Eliz.
 
There are twenty vertices. Take them two at a time. Then remove the sides that you counted.
 
# of sides --- # of diagonals

3 --- 0
4 --- 2
5 --- 5
6 --- 9
7 --- 14

... and so on

see a pattern?
 
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.
 
Well since we have given a complete answer, it might as well be done correctly.

If n>2, we can construct a polygon, by the way it does not have to be convex. So we have n edges and a diagonal is the edge between any two non-adjacent vertices. Joining any two vertices we get a complete graph \(\displaystyle K_n\) which has \(\displaystyle n \choose 2\). That includes the n edges so lets remove them.
\(\displaystyle \begin{array}{rcl}
{n \choose 2} - n & = & \frac{{n\left( {n - 1} \right)}}{2} - n \\
& = & \frac{{n^2 - n - 2n}}{2} \\
& = & \frac{{n\left( {n - 3} \right)}}{2} \\
\end{array}\)
 
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