The length of a wire which is tied as a boundary of a semi ciruclar park is 72 cm. Find the radius and the area of the semi circular park

Answer

To solve the problem, we need to find the radius and the area of a semicircular park given that the length of the wire tied as a boundary is 72 cm.

### Step-by-Step Solution:

1. **Understanding the Perimeter of a Semicircle**:
The perimeter \( P \) of a semicircular park is given by the formula:
\[ P = \pi r + 2r \]
where \( r \) is the radius of the semicircle. The term \( \pi r \) represents the curved part (half the circumference of a full circle), and \( 2r \) is the diameter.

2. **Setting Up the Equation**:
Given that the total length of the wire (which is the perimeter of the semicircle) is 72 cm, we can set up the equation:
\[ \pi r + 2r = 72 \]

3. **Substituting the Value of \(\pi\)**:
For calculations, we can use \(\pi \approx \frac{22}{7}\). Substituting this into the equation gives:
\[ \frac{22}{7} r + 2r = 72 \]

4. **Finding a Common Denominator**:
To combine the terms, we can express \( 2r \) with a common denominator:
\[ 2r = \frac{14}{7} r \]
Thus, the equation becomes:
\[ \frac{22}{7} r + \frac{14}{7} r = 72 \]

5. **Combining the Terms**:
Now, combine the fractions:
\[ \frac{22 + 14}{7} r = 72 \]
This simplifies to:
\[ \frac{36}{7} r = 72 \]

6. **Solving for \( r \)**:
To find \( r \), multiply both sides by \( \frac{7}{36} \):
\[ r = 72 \times \frac{7}{36} \]
Simplifying this gives:
\[ r = 14 \text{ cm} \]

7. **Finding 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 \( r \):
\[ A = \frac{1}{2} \times \frac{22}{7} \times (14)^2 \]

8. **Calculating the Area**:
First, calculate \( (14)^2 = 196 \):
\[ A = \frac{1}{2} \times \frac{22}{7} \times 196 \]
Now simplify:
\[ A = \frac{22 \times 196}{14} \]
Since \( 196 \div 14 = 14 \):
\[ A = 22 \times 14 = 308 \text{ cm}^2 \]

### Final Answers:
- The radius of the semicircular park is **14 cm**.
- The area of the semicircular park is **308 cm²**.