If the area of regular hexagon is `96sqrt(3)`, what is its perimeter?

Answer

To find the perimeter of a regular hexagon given its area, we can follow these steps:

### Step 1: Use the formula for the area of a regular hexagon
The area \( A \) of a regular hexagon with side length \( a \) is given by the formula:
\[ A = \frac{3\sqrt{3}}{2} a^2 \]

### Step 2: Set the area equal to the given area
We know from the problem that the area of the hexagon is \( 96\sqrt{3} \). Therefore, we can set up the equation:
\[ \frac{3\sqrt{3}}{2} a^2 = 96\sqrt{3} \]

### Step 3: Simplify the equation
To eliminate \( \sqrt{3} \) from both sides, we can divide by \( \sqrt{3} \):
\[ \frac{3}{2} a^2 = 96 \]

### Step 4: Solve for \( a^2 \)
Next, we multiply both sides by \( \frac{2}{3} \) to isolate \( a^2 \):
\[ a^2 = 96 \times \frac{2}{3} \]
Calculating the right side:
\[ a^2 = 96 \times \frac{2}{3} = 64 \]

### Step 5: Find \( a \)
Now, we take the square root of both sides to find \( a \):
\[ a = \sqrt{64} = 8 \]

### Step 6: Calculate the perimeter
The perimeter \( P \) of a regular hexagon is given by the formula:
\[ P = 6a \]
Substituting the value of \( a \):
\[ P = 6 \times 8 = 48 \]

### Final Answer
Thus, the perimeter of the regular hexagon is \( 48 \).

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