The perimeter of a semi -circular region is `144 cm `. What is its area ?

Options

A. `1222 cm^(2)`
B. `1234 cm^(2)`
C. `1122 cm^(2)`
D. `1232 cm^(2)`

Answer

To find the area of a semi-circular region given its perimeter, we can follow these steps:

### Step 1: Understand the formula for the perimeter of a semicircle
The perimeter \( P \) of a semicircular region can be expressed as:
\[ P = \pi R + 2R \]
where \( R \) is the radius of the semicircle.

### Step 2: Set up the equation
We know the perimeter is given as \( 144 \, \text{cm} \). Therefore, we can write:
\[ \pi R + 2R = 144 \]

### Step 3: Factor out \( R \)
We can factor \( R \) out of the left side:
\[ R(\pi + 2) = 144 \]

### Step 4: Solve for \( R \)
To find \( R \), we can rearrange the equation:
\[ R = \frac{144}{\pi + 2} \]

### Step 5: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \), we can substitute this value into the equation:
\[ R = \frac{144}{\frac{22}{7} + 2} \]

### Step 6: Simplify the denominator
Convert \( 2 \) into a fraction with a common denominator:
\[ 2 = \frac{14}{7} \]
Thus,
\[ \pi + 2 = \frac{22}{7} + \frac{14}{7} = \frac{36}{7} \]

### Step 7: Substitute back to find \( R \)
Now substitute back into the equation for \( R \):
\[ R = \frac{144}{\frac{36}{7}} = 144 \times \frac{7}{36} \]

### Step 8: Simplify \( R \)
Calculating this gives:
\[ R = \frac{144 \times 7}{36} = \frac{1008}{36} = 28 \, \text{cm} \]

### Step 9: Calculate the area of the semicircle
The area \( A \) of a semicircle is given by:
\[ A = \frac{1}{2} \pi R^2 \]
Substituting \( R = 28 \, \text{cm} \):
\[ A = \frac{1}{2} \times \frac{22}{7} \times (28)^2 \]

### Step 10: Calculate \( (28)^2 \)
Calculating \( (28)^2 \):
\[ (28)^2 = 784 \]

### Step 11: Substitute and simplify the area
Now substitute back into the area formula:
\[ A = \frac{1}{2} \times \frac{22}{7} \times 784 \]
Calculating this gives:
\[ A = \frac{22 \times 784}{14} = \frac{17248}{14} = 1232 \, \text{cm}^2 \]

### Final Answer
Thus, the area of the semicircular region is:
\[ \boxed{1232 \, \text{cm}^2} \]