If a polygon has 54 diagonals, then number of sides in the polygon is

Let's solve this step by step.

Step 1: Recall the formula for the number of diagonals in a polygon. For a polygon with n sides, the number of diagonals can be calculated using the formula:

Number of diagonals = n(n-3)/2

Step 2: We are given that the polygon has 54 diagonals. So, we can set up the equation:

54 = n(n-3)/2

Step 3: Multiply both sides of the equation by 2 to eliminate the fraction:

108 = n(n-3)

Step 4: Expand the equation:

108 = n^2 - 3n

Step 5: Rearrange the equation to form a quadratic equation:

n^2 - 3n - 108 = 0

Step 6: Factorize the quadratic equation:

(n - 12)(n + 9) = 0

Step 7: Set each factor equal to zero and solve for n:

n - 12 = 0 or n + 9 = 0

n = 12 or n = -9

Step 8: Since the number of sides in a polygon cannot be negative, we discard the solution n = -9.

Step 9: Therefore, the number of sides in the polygon is 12.

So, the number of sides in the polygon is 12.