Understanding Polygons and Diagonals
To find the number of sides in a polygon given the number of diagonals, we can use the formula for the number of diagonals in a polygon with 'n' sides:
Diagonals Formula
- The formula is: D = n(n - 3) / 2
- Here, D represents the number of diagonals, and n is the number of sides.
Setting Up the Equation
Given that the polygon has 27 diagonals, we can set up the equation:
- 27 = n(n - 3) / 2
Solving for n
To eliminate the fraction, multiply both sides by 2:
- 54 = n(n - 3)
Next, rearranging gives us:
- n(n - 3) - 54 = 0
- n^2 - 3n - 54 = 0
Factoring the Quadratic
Now, we need to factor the quadratic equation:
- (n - 9)(n + 6) = 0
This gives us two potential solutions for n:
- n - 9 = 0 → n = 9
- n + 6 = 0 → n = -6 (not valid since sides cannot be negative)
Conclusion
Thus, the polygon with 27 diagonals has:
- 9 sides
This method effectively illustrates how to determine the number of sides in a polygon based on the number of diagonals.