If the perimeter of a semi-circle is 108 cm, then find its radius (in cm).

Options

A. 42
B. 28
C. 56
D. 21

Answer

To find the radius 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 can be expressed as:
\[ P = \pi r + d \]
where \( d \) is the diameter of the semicircle.

### Step 2: Relate the diameter to the radius
The diameter \( d \) of a semicircle is twice the radius:
\[ d = 2r \]
Substituting this into the perimeter formula gives:
\[ P = \pi r + 2r \]

### Step 3: Combine the terms
We can factor out \( r \) from the equation:
\[ P = r(\pi + 2) \]

### Step 4: Substitute the given perimeter
We know from the problem that the perimeter \( P \) is 108 cm. Therefore, we can set up the equation:
\[ 108 = r(\pi + 2) \]

### Step 5: Solve for \( r \)
To isolate \( r \), we rearrange the equation:
\[ r = \frac{108}{\pi + 2} \]

### Step 6: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \), we can substitute this value into the equation:
\[ \pi + 2 = \frac{22}{7} + 2 = \frac{22}{7} + \frac{14}{7} = \frac{36}{7} \]

### Step 7: Substitute back to find \( r \)
Now substitute \( \pi + 2 \) back into the equation for \( r \):
\[ r = \frac{108}{\frac{36}{7}} = 108 \times \frac{7}{36} \]

### Step 8: Simplify the expression
Calculating this gives:
\[ r = \frac{108 \times 7}{36} = \frac{756}{36} = 21 \]

### Conclusion
Thus, the radius \( r \) of the semicircle is:
\[ \boxed{21 \text{ cm}} \]