If the perimeter of a semicircle is 72 cm, then find its area (in `cm^(2)` ).

Options

A. 308
B. 616
C. 160
D. 320

Answer

To solve the problem, we will follow these steps:

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

### Step 2: Set up the equation with the given perimeter
Given that the perimeter of the semicircle is 72 cm, we can set up the equation:
\[ \pi r + 2r = 72 \]

### Step 3: Factor out the radius
We can factor out \( r \) from the left side of the equation:
\[ r(\pi + 2) = 72 \]

### Step 4: Solve for the radius \( r \)
To find \( r \), we can rearrange the equation:
\[ r = \frac{72}{\pi + 2} \]

### Step 5: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \) for calculation:
\[ r = \frac{72}{\frac{22}{7} + 2} \]
First, convert 2 into a fraction:
\[ 2 = \frac{14}{7} \]
Now, add the fractions:
\[ \frac{22}{7} + \frac{14}{7} = \frac{36}{7} \]
Now substitute this back into the equation for \( r \):
\[ r = \frac{72}{\frac{36}{7}} = 72 \times \frac{7}{36} = 14 \text{ cm} \]

### Step 6: 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 the value of \( r \):
\[ A = \frac{1}{2} \pi (14)^2 \]
\[ A = \frac{1}{2} \pi (196) \]
\[ A = 98\pi \]

### Step 7: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ A = 98 \times \frac{22}{7} \]
\[ A = \frac{2156}{7} \]
Calculating this gives:
\[ A \approx 308 \text{ cm}^2 \]

### Final Answer
The area of the semicircle is approximately \( 308 \text{ cm}^2 \).

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