Does a polygon ever have more diagonals than sides?

Answer 1

The number of diagonals for a shape with x sides is (x²-3x)/2

The number of sides is x

You want to know if x is ever less than (x²-3x)/2 or else (x²-3x)/2 - x > 0

(x² - 3x) / 2 - x > 0

(x² - 3x)/2 - (2x)/x > 0

x² - 3x - 2x > 0

x² - 5x > 0

x(x-5)>0

This only true if x<0 or x>5, and since a polygon cannot have less than 2 sides, x<0 is meaningless.

This means that if x<5 then the number of diagonals is less than the number of sides, equal for x=5, and more for x>5. ■