The angle substended by an arc at the centre of a circle is `70^(@)` . If the circumference of the circle is 132 cm , then find the area of the sector formed .

Options

A. `269 .5 cm^(2)`
B. `1078 cm^(2)`
C. `539 cm^(2)`
D. `1617 cm^(2)`

Answer

To find the area of the sector formed by an arc that subtends an angle of \(70^\circ\) at the center of a circle with a circumference of \(132 \, \text{cm}\), we can follow these steps:

### Step 1: Find the radius of the circle
The formula for the circumference of a circle is given by:
\[ C = 2\pi R \]
where \(C\) is the circumference and \(R\) is the radius. Given that the circumference is \(132 \, \text{cm}\), we can set up the equation:
\[ 2\pi R = 132 \]
To find \(R\), we can rearrange the equation:
\[ R = \frac{132}{2\pi} = \frac{66}{\pi} \]

### Step 2: Substitute the value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\):
\[ R = \frac{66}{\frac{22}{7}} = 66 \times \frac{7}{22} = 3 \times 7 = 21 \, \text{cm} \]

### Step 3: Use the formula for the area of the sector
The area \(A\) of a sector of a circle is given by the formula:
\[ A = \frac{\theta}{360} \times \pi R^2 \]
where \(\theta\) is the angle in degrees. In this case, \(\theta = 70^\circ\) and \(R = 21 \, \text{cm}\).

### Step 4: Substitute the values into the area formula
Substituting the values into the formula:
\[ A = \frac{70}{360} \times \pi \times (21)^2 \]

### Step 5: Simplify the expression
Calculating \(21^2\):
\[ 21^2 = 441 \]
Now substituting this back into the area formula:
\[ A = \frac{70}{360} \times \pi \times 441 \]

### Step 6: Simplify the fraction
We can simplify \(\frac{70}{360}\):
\[ \frac{70}{360} = \frac{7}{36} \]
Thus, the area becomes:
\[ A = \frac{7}{36} \times \pi \times 441 \]

### Step 7: Calculate the area
Using \(\pi \approx \frac{22}{7}\):
\[ A = \frac{7}{36} \times \frac{22}{7} \times 441 \]
The \(7\) cancels out:
\[ A = \frac{22}{36} \times 441 \]
Now simplify \(\frac{22}{36} = \frac{11}{18}\):
\[ A = \frac{11}{18} \times 441 \]

### Step 8: Calculate \(441 \div 18\)
Calculating \(441 \div 18\):
\[ 441 \div 18 = 24.5 \]
Now multiplying by \(11\):
\[ A = 11 \times 24.5 = 269.5 \, \text{cm}^2 \]

### Final Answer
The area of the sector formed is:
\[ \boxed{269.5 \, \text{cm}^2} \]