A metal wire, when bent in the form of an equilateral triangle of largest area, enclosed 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 circle.

Answer

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

### Step 1: Understand the area of the equilateral triangle
We are given that the area of the equilateral triangle is \( 484\sqrt{3} \, \text{cm}^2 \).

### Step 2: Use the formula for the area of an equilateral triangle
The formula for the area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
We can set this equal to the given area:
\[ \frac{\sqrt{3}}{4} a^2 = 484\sqrt{3} \]

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

### Step 4: Find the side length \( a \)
Taking the square root of both sides gives us:
\[ a = \sqrt{1936} = 44 \, \text{cm} \]

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

### Step 6: Relate the perimeter of the triangle to the circumference of the circle
The perimeter of the triangle is equal to the circumference \( C \) of the circle formed by the same wire:
\[ C = 2\pi r \]
Thus, we have:
\[ 132 = 2\pi r \]

### Step 7: Solve for the radius \( r \)
Rearranging the equation to solve for \( r \):
\[ r = \frac{132}{2\pi} = \frac{66}{\pi} \]

### Step 8: Calculate the area of the circle
The area \( A \) of the circle is given by:
\[ A = \pi r^2 \]
Substituting for \( r \):
\[ A = \pi \left(\frac{66}{\pi}\right)^2 = \pi \cdot \frac{4356}{\pi^2} = \frac{4356}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ A = \frac{4356 \times 7}{22} = 1386 \, \text{cm}^2 \]

### Final Answer:
The area of the circle is \( 1386 \, \text{cm}^2 \).

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