(i) a hexagon
As 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.
Therefore number of lines formed = 6C2
On using the formula,
nCr = n!/r!(n – r)!
6C2 = 6!/2!(6 - 2)!
= 6! / (2! 4!)
= [6 × 5 × 4!] / (2! 4!)
= [6 × 5] / (2 × 1)
= 3 × 5
= 15
As we know that number of sides of hexagon is 6
Therefore, number of diagonals = 15 – 6 = 9
Total no. of diagonals formed is 9.
(ii) a polygon of 16 sides
As 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.
Therefore number of lines formed = 16C2
On using the formula,
nCr = n!/r!(n – r)!
16C2 = 16!/2!(16 - 2)!
= 16! / (2! 14!)
= [16 × 15 × 14!] / (2! 14!)
= [16 × 15] / (2 × 1)
= 8 × 15
= 120
As we know that number of sides of a polygon is 16
Therefore, number of diagonals = 120 – 16 = 104
Total no. of diagonals formed is 104.