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Defining a diagonal as any two vertices that are not connected by an edge (non-adjacent) of the polygon, you can determine the amount of diagonals using the following formula

s = sides in polygon

(s*(s-3))/2

So,

(3*(3-3))/2 = 0

(4*(4-3))/2 = 2

(5*(5-3))/2 = 5

(6*(6-3))/2 = 9

(7*(7-3))/2 = 14

(8*(8-3))/2 = 20

(9*(9-3))/2 = 27

(10*(10-3))/2 = 35

(11*(11-3))/2 = 44

So, the polygon would have to have somewhere between 10 and 11 sides. Since a polygon can not have fractional amount of sides, no polygon exists that has 42 diagonals.

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Q: How many sides would you find in a polygon which has 42 diagonals?
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