The perimeter of a certain sector of a circle of radius 5.7 m is 27.2 m. find the area of the sector

Answer

To solve the problem, we need to find the area of the sector of a circle given its perimeter. Here's a step-by-step solution:

### Step 1: Understand the Perimeter of the Sector
The perimeter \( P \) of a sector of a circle can be calculated using the formula:
\[ P = 2r + l \]
where \( r \) is the radius of the circle and \( l \) is the length of the arc of the sector.

### Step 2: Write the Formula for the Length of the Arc
The length of the arc \( l \) can be expressed as:
\[ l = \frac{\theta}{360} \times 2\pi r \]
where \( \theta \) is the angle of the sector in degrees.

### Step 3: Substitute Known Values into the Perimeter Formula
Given:
- Radius \( r = 5.7 \) m
- Perimeter \( P = 27.2 \) m

We can substitute these values into the perimeter formula:
\[ 27.2 = 2(5.7) + \frac{\theta}{360} \times 2\pi(5.7) \]

### Step 4: Calculate \( 2r \)
Calculating \( 2r \):
\[ 2(5.7) = 11.4 \text{ m} \]

### Step 5: Substitute and Rearrange the Equation
Substituting \( 2r \) back into the equation:
\[ 27.2 = 11.4 + \frac{\theta}{360} \times 2\pi(5.7) \]
Now, isolate the arc length term:
\[ 27.2 - 11.4 = \frac{\theta}{360} \times 2\pi(5.7) \]
\[ 15.8 = \frac{\theta}{360} \times 2\pi(5.7) \]

### Step 6: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = \frac{15.8 \times 360}{2\pi(5.7)} \]
Calculating \( 2\pi(5.7) \):
\[ 2\pi(5.7) \approx 35.796 \]
Thus,
\[ \theta = \frac{15.8 \times 360}{35.796} \approx 158.9^\circ \]

### 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:
\[ A = \frac{158.9}{360} \times \pi(5.7)^2 \]
Calculating \( \pi(5.7)^2 \):
\[ \pi(5.7)^2 \approx 3.14 \times 32.49 \approx 102.24 \]
Thus,
\[ A = \frac{158.9}{360} \times 102.24 \approx 45.03 \text{ m}^2 \]

### Final Answer
The area of the sector is approximately \( 45.03 \, \text{m}^2 \).

---