Find the number of diagonals of

(i) a hexagon

We know that a hexagon has 6 angular points. By joining those any two angular points we get a line which is either a side or a diagonal.

So number of lines formed = ⁶C₂

By using the formula,

ⁿC_r = \( \dfrac{ n!}{r!(n – r)!}\)

⁶C₂ =\( \dfrac{6!}{2!(6-2)!}\)

=\(\dfrac{ 6×5 }{ (2×1)}\)

= 3×5

= 15

We know number of sides of hexagon is 6

So, number of diagonals = 15 – 6 = 9

The total no. of diagonals formed is 9.

(ii) a polygon of 16 sides

We know that a polygon of 16 sides has 16 angular points. By joining those any two angular points we get a line which is either a side or a diagonal.

So number of lines formed = ¹⁶C₂

By using the formula,

ⁿC_r =\( \dfrac{ n!}{r!(n – r)!}\)

¹⁶C₂ = \(\dfrac{ 16!}{2!(16-2)!}\)

= \(\dfrac{ 16×15 }{ (2×1)}\)

= 8×15

= 120

We know number of sides of a polygon is 16

So, number of diagonals = 120 – 16 = 104

The total no. of diagonals formed is 104.