How many sides are there in a polygon that has 20 diagonals?

Calculation:

Given,

Number of diagonals, D = 20 

Formula for the number of diagonals in a polygon is:

\(D = \frac{n(n - 3)}{2}\)

Substitute the value of D = 20  in the formula:

⇒ \(20 = \frac{n(n - 3)}{2}\)

Multiply both sides by 2:

⇒ \(40 = n(n - 3)\)

⇒ \(n^2 - 3n - 40 = 0\)

We now solve this quadratic equation:

\(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

⇒ \(n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-40)}}{2(1)}\)

⇒ \(n = \frac{3 \pm 13}{2}\)

The two solutions are:

⇒ \(n = \frac{3 + 13}{2} = 8\)

or

⇒ \(n = \frac{3 - 13}{2} = -5\)

Since the number of sides cannot be negative, we choose n = 8.

∴ The polygon has 8 sides.

Hence, the correct answer is Option 3.