A wire is bent into the form of a circle, whose area is `154 cm^(2)`. If the same wire is bent into the form of an equilateral triangle, the approximate area of the equilateral triangle is

Options

A. `93.14 cm^(2)`
B. `90.14 cm^(2)`
C. `83.14 cm^(2)`
D. `39.14 cm^(2)`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Find the radius of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle. We know the area is \( 154 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 154 = \pi r^2 \]

### Step 2: Solve for \( r^2 \)
To find \( r^2 \), we can rearrange the equation:
\[ r^2 = \frac{154}{\pi} \]
Using \( \pi \approx 3.14 \):
\[ r^2 \approx \frac{154}{3.14} \approx 49.04 \]

### Step 3: Calculate the radius \( r \)
Now, we take the square root of \( r^2 \):
\[ r \approx \sqrt{49.04} \approx 7 \, \text{cm} \]

### Step 4: Find the circumference of the circle
The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Substituting the value of \( r \):
\[ C \approx 2 \times 3.14 \times 7 \approx 43.96 \, \text{cm} \]

### Step 5: Find the side length of the equilateral triangle
When the wire is bent into the form of an equilateral triangle, the perimeter of the triangle is equal to the circumference of the circle. Therefore, the perimeter \( P \) of the equilateral triangle is:
\[ P = 3s = C \]
where \( s \) is the side length of the triangle. Thus:
\[ 3s \approx 43.96 \]
Solving for \( s \):
\[ s \approx \frac{43.96}{3} \approx 14.65 \, \text{cm} \]

### Step 6: Calculate the area of the equilateral triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting the value of \( s \):
\[ A \approx \frac{\sqrt{3}}{4} (14.65)^2 \]
Calculating \( (14.65)^2 \):
\[ (14.65)^2 \approx 214.6225 \]
Now substituting this back into the area formula:
\[ A \approx \frac{\sqrt{3}}{4} \times 214.6225 \approx 0.433 \times 214.6225 \approx 93.14 \, \text{cm}^2 \]

### Final Answer
The approximate area of the equilateral triangle is \( 93.14 \, \text{cm}^2 \).

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