If the perimeter of a semicircle is 108 cm, then find its area (in `cm^2` ).

Options

A. A) 1386
B. B) 512
C. C) 693
D. D) 1024

Answer

To find the area of a semicircle given its perimeter, we can follow these steps:

### Step 1: Understand the formula for the perimeter of a semicircle
The perimeter \( P \) of a semicircle is given by the formula:
\[ P = \pi r + 2r \]
where \( r \) is the radius of the semicircle.

### Step 2: Set up the equation using the given perimeter
We know from the problem that the perimeter is 108 cm. Therefore, we can set up the equation:
\[ \pi r + 2r = 108 \]

### Step 3: Factor out the radius \( r \)
We can factor \( r \) out of the left side of the equation:
\[ r(\pi + 2) = 108 \]

### Step 4: Solve for the radius \( r \)
To find \( r \), we divide both sides by \( \pi + 2 \):
\[ r = \frac{108}{\pi + 2} \]
Using \( \pi \approx \frac{22}{7} \), we can substitute this value in:
\[ r = \frac{108}{\frac{22}{7} + 2} \]
First, convert 2 to a fraction with a denominator of 7:
\[ 2 = \frac{14}{7} \]
Thus,
\[ r = \frac{108}{\frac{22 + 14}{7}} = \frac{108 \times 7}{36} = \frac{756}{36} = 21 \text{ cm} \]

### Step 5: Calculate the area of the semicircle
The area \( A \) of a semicircle is given by the formula:
\[ A = \frac{1}{2} \pi r^2 \]
Substituting \( r = 21 \) cm into the formula:
\[ A = \frac{1}{2} \times \frac{22}{7} \times (21)^2 \]
Calculating \( 21^2 \):
\[ 21^2 = 441 \]
Now substituting this back into the area formula:
\[ A = \frac{1}{2} \times \frac{22}{7} \times 441 \]
Calculating \( \frac{22 \times 441}{14} \):
\[ A = \frac{9702}{14} = 693 \text{ cm}^2 \]

### Final Answer
The area of the semicircle is \( 693 \text{ cm}^2 \).

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