If the area of a semi-circle be 77 sq m, find its perimeter.

Options

A. 36 m
B. 4 m
C. 48 m
D. 52 m

Answer

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

### Step 1: Understand the formula for 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 using the given area.
We know from the problem that the area of the semicircle is 77 sq m. 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 can substitute this into our equation:
\[ \frac{1}{2} \times \frac{22}{7} \times r^2 = 77 \]

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

### 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:
\[ r = \sqrt{49} = 7 \text{ m} \]

### 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 \( r \) and \( \pi \):
\[ P = \frac{22}{7} \times 7 + 2 \times 7 \]

### Step 8: Simplify the expression for the perimeter.
Calculating the first term:
\[ P = 22 + 14 = 36 \text{ m} \]

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

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