The perimeter of a sector of a circle of radius 5.6 cm is 27.2 cm. Find the area of the sector.

Answer

To solve the problem of finding the area of a sector of a circle with a given radius and perimeter, we can follow these steps:

### Step 1: Understand the given information
We are given:
- Radius (r) = 5.6 cm
- Perimeter of the sector = 27.2 cm

### Step 2: Write the formula for the perimeter of a sector
The perimeter (P) of a sector of a circle can be expressed as:
\[ P = r + r + L \]
Where \( L \) is the length of the arc of the sector. Therefore, we can rewrite it as:
\[ P = 2r + L \]

### Step 3: Substitute the known values into the perimeter formula
Substituting the known values into the perimeter formula:
\[ 27.2 = 2(5.6) + L \]

### Step 4: Calculate the length of the arc (L)
First, calculate \( 2(5.6) \):
\[ 2(5.6) = 11.2 \]
Now, substitute this back into the equation:
\[ 27.2 = 11.2 + L \]
To find \( L \), subtract 11.2 from both sides:
\[ L = 27.2 - 11.2 = 16 \text{ cm} \]

### Step 5: Use the arc length to find the angle (θ) of the sector
The length of the arc (L) can also be expressed in terms of the angle (θ) in degrees:
\[ L = \frac{\theta}{360} \times 2\pi r \]
Substituting the known values:
\[ 16 = \frac{\theta}{360} \times 2\pi(5.6) \]

### Step 6: Solve for θ
First, calculate \( 2\pi(5.6) \):
Using \( \pi \approx \frac{22}{7} \):
\[ 2\pi(5.6) = 2 \times \frac{22}{7} \times 5.6 = \frac{44 \times 5.6}{7} = \frac{246.4}{7} \approx 35.2 \]
Now substitute back:
\[ 16 = \frac{\theta}{360} \times 35.2 \]
To isolate θ, multiply both sides by 360:
\[ 16 \times 360 = \theta \times 35.2 \]
\[ 5760 = \theta \times 35.2 \]
Now divide both sides by 35.2:
\[ \theta = \frac{5760}{35.2} \approx 163.64 \text{ degrees} \]

### Step 7: Calculate the area of the sector
The area (A) of the sector can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Substituting the values:
\[ A = \frac{163.64}{360} \times \pi(5.6)^2 \]
Calculating \( (5.6)^2 = 31.36 \):
\[ A = \frac{163.64}{360} \times \frac{22}{7} \times 31.36 \]

### Step 8: Calculate the area
First, calculate \( \frac{163.64 \times 22 \times 31.36}{360 \times 7} \):
\[ A \approx \frac{163.64 \times 22 \times 31.36}{2520} \]
Calculating the numerator:
\[ 163.64 \times 22 \times 31.36 \approx 11300.8 \]
Now divide by 2520:
\[ A \approx \frac{11300.8}{2520} \approx 4.48 \text{ cm}^2 \]

### Final Answer
The area of the sector is approximately \( 44.8 \text{ cm}^2 \).

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