How many diagonals are there in a polygon of 20 sides?

Answer 1

20 sides implies 20 corners.
To ask how many ways there are of connecting each corner with _every_ other corner is the same as asking how many ways you can choose two items from 20. There is a formula for calculating that number. However, we must omit 20 of these ways because are the sides of the polygon, not diagonals.


Hence, the number we seek is "20 choose 2" - 20 = 190 - 20 = 170


To calculate "20 choose 2" easily use

* * * * *

Convincing but wrong. Joining a vertex to either of the two adjoining vertices gives a side of the polygon, NOT a diagonal. So the number we seek is NOT 20 choose 2. What we want is 20 seek any of the 20-3 vertices (not itself and not the two adjacent ones). Next, the suggested method would count each diaginal twice - once from each end. So we need to divide the result by 2.

The correct answer for an n-gon is 0.5*n*(n-3)
When n = 20 this gives 0.5*20*17 = 170