The perimeter of a semicircular plate is 108 cm, find its area.

Answer

To find the area of a semicircular plate with a given perimeter of 108 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Perimeter of a Semicircle**:
The perimeter \( P \) of a semicircular plate is given by the formula:
\[ P = \pi r + 2r \]
where \( r \) is the radius of the semicircle.

2. **Set Up the Equation**:
We know from the problem that the perimeter is 108 cm. Therefore, we can set up the equation:
\[ \pi r + 2r = 108 \]

3. **Factor Out \( r \)**:
We can factor \( r \) out of the left side:
\[ r(\pi + 2) = 108 \]

4. **Solve for \( r \)**:
To find \( r \), we divide both sides by \( \pi + 2 \):
\[ r = \frac{108}{\pi + 2} \]

5. **Substitute the Value of \( \pi \)**:
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{108}{\frac{22}{7} + 2} \]
To simplify \( \frac{22}{7} + 2 \), convert 2 into a fraction:
\[ 2 = \frac{14}{7} \]
So,
\[ \frac{22}{7} + \frac{14}{7} = \frac{36}{7} \]
Therefore,
\[ r = \frac{108 \times 7}{36} = \frac{756}{36} = 21 \text{ cm} \]

6. **Calculate the Area of the Semicircle**:
The area \( A \) of a semicircle is given by:
\[ A = \frac{1}{2} \pi r^2 \]
Substituting \( r = 21 \) cm:
\[ A = \frac{1}{2} \times \frac{22}{7} \times (21)^2 \]
Calculate \( (21)^2 = 441 \):
\[ A = \frac{1}{2} \times \frac{22}{7} \times 441 \]

7. **Simplify the Area Calculation**:
\[ A = \frac{22 \times 441}{14} = \frac{9702}{14} = 693 \text{ cm}^2 \]

### Final Answer:
The area of the semicircular plate is \( 693 \text{ cm}^2 \).

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