The area of a semicircle is `77 cm^(2)`. Calculate its perimeter (in cm).

Options

A. 72
B. 49
C. 98
D. 36

Answer

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

### Step 1: Understand the area of a semicircle
The area \( A \) of a semicircle is given by the formula:
\[ A = \frac{1}{2} \pi r^2 \]
where \( r \) is the radius of the semicircle.

### Step 2: Set up the equation with the given area
We know that the area of the semicircle is \( 77 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ \frac{1}{2} \pi r^2 = 77 \]

### Step 3: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \), we substitute this value into the equation:
\[ \frac{1}{2} \times \frac{22}{7} \times r^2 = 77 \]

### Step 4: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ \frac{22}{7} r^2 = 154 \]

### Step 5: Solve for \( r^2 \)
Now, multiply both sides by \( \frac{7}{22} \):
\[ r^2 = 154 \times \frac{7}{22} \]
Calculating the right side:
\[ r^2 = 154 \times \frac{7}{22} = 154 \div 22 \times 7 = 7 \times 7 = 49 \]

### Step 6: Find the radius \( r \)
Taking the square root of both sides gives us:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Step 7: Calculate the perimeter of the semicircle
The perimeter \( P \) of a semicircle is given by the formula:
\[ P = \pi r + 2r \]
Substituting the value of \( r \):
\[ P = \frac{22}{7} \times 7 + 2 \times 7 \]
Calculating this:
\[ P = 22 + 14 = 36 \, \text{cm} \]

### Final Answer
The perimeter of the semicircle is \( 36 \, \text{cm} \).

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