Question

A polygon has 27 diagonals. How many sides does it have??? ​

Answer 1

Number of sides in a polygon = n(n - 3)/2

Here, n = sides of the polygon.

Given :-

Diagonals = 27

To find :-

Number of sides.

Solution :-

n(n - 3)/2 = 27

n² - 3n = 27 x 2

n² - 3n = 54

n² - 3n - 54 = 0

Solving for n :-

n² - 3n - 54 = 0

(n − 9)(n + 6)  = 0

n = 9 or n = -6

Number of sides cannot be negative, therefore there are 9 sides in the polygon.

Answer 2

Given,

Number of diagonals in a polygon = 27

To find,

The number of sides of the polygon.

Solution,

We can simply solve this mathematical problem using the following process:

Let us assume that the polygon has "n" number of sides.

For any polygon containing "n" number of sides, the total number of diagonals

= n(n-3)/2 {Equation-1}

Now, as per the question;

Total number of diagonals in the polygon = 27

=> n(n-3)/2 = 27 {Using equation-1}

=> n^2 - 3n = 54

=> n^2 - 3n - 54 = 0

=> n^2 - 9n + 6n - 54 = 0

=> n(n-9) + 6(n-9) = 0

=> (n-9)(n+6) = 0

=> (n-9) = 0 or (n+6)=0

=> n = 9 or n = -6

But any polygon cannot have a negative number of sides. Thus, n = -6 is invalid in this case and 9 becomes the only possible value of n.

Hence, the polygon has 9 sides.