A triangle and a regular hexagon have the same perimeter. If the area of the hexagon is `72 sqrt(3)`, what is the area of the triangle ?

Options

A. `62.354`
B. `83.138`
C. `101.823`
D. `103.923`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Area of the Hexagon
The area \( A \) of a regular hexagon can be calculated using the formula:
\[ A = \frac{3\sqrt{3}}{2} s^2 \]
where \( s \) is the length of a side of the hexagon.

### Step 2: Set Up the Equation
Given that the area of the hexagon is \( 72\sqrt{3} \), we can set up the equation:
\[ \frac{3\sqrt{3}}{2} s^2 = 72\sqrt{3} \]

### Step 3: Simplify the Equation
To simplify, we can divide both sides by \( \sqrt{3} \):
\[ \frac{3}{2} s^2 = 72 \]

### Step 4: Solve for \( s^2 \)
Next, we multiply both sides by \( \frac{2}{3} \):
\[ s^2 = 72 \cdot \frac{2}{3} = 48 \]

### Step 5: Find \( s \)
Now, take the square root of both sides to find \( s \):
\[ s = \sqrt{48} = 4\sqrt{3} \]

### Step 6: Calculate the Perimeter of the Hexagon
The perimeter \( P \) of the hexagon is given by:
\[ P = 6s = 6 \cdot 4\sqrt{3} = 24\sqrt{3} \]

### Step 7: Find the Side Length of the Triangle
Since the triangle has the same perimeter as the hexagon, the perimeter of the triangle is also \( 24\sqrt{3} \). The perimeter \( P \) of a triangle with side length \( a \) is given by:
\[ P = 3a \]
Setting this equal to the perimeter of the hexagon:
\[ 3a = 24\sqrt{3} \]
Solving for \( a \):
\[ a = \frac{24\sqrt{3}}{3} = 8\sqrt{3} \]

### Step 8: Calculate the Area of the Triangle
Now, we can find the area \( A \) of the triangle using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = 8\sqrt{3} \):
\[ A = \frac{\sqrt{3}}{4} (8\sqrt{3})^2 \]
Calculating \( (8\sqrt{3})^2 \):
\[ (8\sqrt{3})^2 = 64 \cdot 3 = 192 \]
Now substituting back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \cdot 192 = 48\sqrt{3} \]

### Final Answer
Thus, the area of the triangle is \( 48\sqrt{3} \).

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