Question

If a polygon has 24 sides, how many diagonals are there from each vertex? How many diagonals are there in all?

Answer 1

324 diagonals

Explanation:

if polygon is

of 4 sides, diagonals=2,diagonal from each vertex=1

of 5 sides, diagonals=5,diagonals from each vertex=2

of 6 sides, diagonals=9,diagonals from each vertex=3

of 7 sides, diagonals=14,diagonals from each vertex=4
`:`
`:`
`:`
`:`
of n sides, diagonals=?,diagonal from each vertex=`n`-3

we can see the series here

`2,5,9,14.........n`th `term`

`n`th term can be found as

1st term`=2=(1^2+3(1))/2`

2nd term`=5=(2^2+3(2))/2`

3rd term`=9=(3^2+3(3))/2`
`:`
`:`
`:`
`:` `n`th term=`(n^2+3n)/2`

(there's a kinda shortcut for that if u'r interested ,contact me on my email)

for polygon with 24 sides put n=24

no. of diagonals are=`(24^2+3(24))/2 = 324`,number of diagonals from each vertex=21