Find the numbers of sides of a polygon whose number of diagonals is
27

Answer

To find the number of sides of a polygon whose number of diagonals is 27, we can use the formula for the number of diagonals in a polygon, which is given by:

\[ D = \frac{n(n - 3)}{2} \]

where \(D\) is the number of diagonals and \(n\) is the number of sides of the polygon.

### Step 1: Set up the equation
We know that the number of diagonals \(D\) is equal to 27. Therefore, we can set up the equation:

\[ \frac{n(n - 3)}{2} = 27 \]

### Step 2: Eliminate the fraction
To eliminate the fraction, we can multiply both sides of the equation by 2:

\[ n(n - 3) = 54 \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to form a standard quadratic equation:

\[ n^2 - 3n - 54 = 0 \]

### Step 4: Factor the quadratic equation
Now we need to factor the quadratic equation. We are looking for two numbers that multiply to -54 (the constant term) and add up to -3 (the coefficient of \(n\)). The numbers -9 and +6 work because:

\[ -9 \times 6 = -54 \quad \text{and} \quad -9 + 6 = -3 \]

Thus, we can rewrite the equation as:

\[ (n - 9)(n + 6) = 0 \]

### Step 5: Solve for \(n\)
Now we can set each factor equal to zero:

1. \(n - 9 = 0 \implies n = 9\)
2. \(n + 6 = 0 \implies n = -6\)

Since the number of sides of a polygon cannot be negative, we discard \(n = -6\).

### Conclusion
Thus, the number of sides of the polygon is:

\[ \boxed{9} \]