Question 1202866: How many diagonals does a polygon with 20 sides have?
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! Draw several of the first few as regular polygons, draw and count the number of diagonals in each, and find the rule which answers your question.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Answer: 170
Reason:
Let n be positive integer such that  .
The value of n can be selected from this set {3, 4, 5, 6, ...}
Consider a convex polygon with n sides.
Select a vertex at random. That particular vertex connects to n-3 other vertices to form n-3 diagonals.
We subtract off 3 because we exclude the selected vertex itself and its two adjacent neighbors.
Do this for all n vertices to have n(n-3) diagonals.
But wait, we've double counted things so we must divide by 2.
The true count of diagonals will be n(n-3)/2
I recommend testing that formula with shapes like squares, pentagons and hexagons.
I also recommend to draw the accompanying diagram to help cement your understanding.
Take careful note that the expression n(n-3) will always be even. The proof is left for the reader.
For a polygon with n = 20 sides there are
n(n-3)/2 = 20*(20-3)/2 = 170 different diagonals
The diagram for this would be very messy to draw out, so I don't recommend it.
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