The number of diagonals of an n-sided regular polygon is 665. What is the value of ‘n’?

The formula for the number of diagonals of an $n$-sided polygon is:

$ \text{Diagonals} = \frac{n(n-3)}{2} $

We are given:

$ \frac{n(n-3)}{2} = 665 $

Multiply both sides by 2:

$ n(n-3) = 1330 $

$ n^2 - 3n - 1330 = 0 $

Step 1: Solve the quadratic equation

Discriminant:

$ D = (-3)^2 - 4 \cdot 1 \cdot (-1330) = 9 + 5320 = 5329 $

$ n = \frac{3 \pm \sqrt{5329}}{2} $

$ \sqrt{5329} = 73 $

$ n = \frac{3 + 73}{2} = \frac{76}{2} = 38 $

(The negative root is not valid.)

✅ Answer: 38