A metal wire, when bent in the form of an equilateral triangle of largest area, encloses an area of `484sqrt(3) cm^(2)`. If the same wire is bent into the form of a circle of largest area, find the area of this circle.

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the area of the equilateral triangle
The area \( A \) of an equilateral triangle with side length \( x \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} x^2 \]
We know from the problem that the area of the triangle is \( 484 \sqrt{3} \, \text{cm}^2 \).

### Step 2: Set up the equation
We can set up the equation using the area formula:
\[ \frac{\sqrt{3}}{4} x^2 = 484 \sqrt{3} \]

### Step 3: Solve for \( x^2 \)
To eliminate \( \sqrt{3} \) from both sides, we divide both sides by \( \sqrt{3} \):
\[ \frac{1}{4} x^2 = 484 \]
Now, multiply both sides by 4:
\[ x^2 = 484 \times 4 \]
Calculating \( 484 \times 4 \):
\[ x^2 = 1936 \]

### Step 4: Find \( x \)
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{1936} = 44 \, \text{cm} \]

### Step 5: Calculate the perimeter of the equilateral triangle
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3x \]
Substituting the value of \( x \):
\[ P = 3 \times 44 = 132 \, \text{cm} \]

### Step 6: Relate the perimeter to the circle
When the wire is bent into the shape of a circle, the circumference \( C \) of the circle is equal to the perimeter of the triangle:
\[ C = 132 \, \text{cm} \]
The circumference of a circle is given by:
\[ C = 2\pi r \]
Setting the two expressions for circumference equal gives:
\[ 2\pi r = 132 \]

### Step 7: Solve for \( r \)
To find the radius \( r \), we can rearrange the equation:
\[ r = \frac{132}{2\pi} \]
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ r = \frac{132}{2 \times \frac{22}{7}} = \frac{132 \times 7}{44} = \frac{924}{44} = 21 \, \text{cm} \]

### Step 8: Calculate the area of the circle
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi \times (21)^2 = \pi \times 441 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{22}{7} \times 441 \]
Calculating:
\[ A = \frac{22 \times 441}{7} = \frac{9702}{7} = 1386 \, \text{cm}^2 \]

### Final Answer
The area of the circle is:
\[ \boxed{1386 \, \text{cm}^2} \]

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