UPSC NDA I 2025 Mathematics Paper

Given,
Number of diagonals, D=20
Formula for the number of diagonals in a polygon is:
D=‌

n(n−3)

2

Substitute the value of D=20 in the formula:
⇒20=‌

n(n−3)

2

Multiply both sides by 2 :
‌⇒40=n(n−3)
‌⇒n2−3n−40=0
We now solve this quadratic equation:
‌n=‌

−b±√b2−4ac

2a

‌⇒n=‌

−(−3)±√(−3)2−4(1)(−40)

2(1)

⇒n=‌

3±13

2

The two solutions are:
⇒n=‌

3+13

2

=8
or
⇒n=‌

3−13

2

=−5
Since the number of sides cannot be negative, we choose n=8.
∴ The polygon has 8 sides.