The perimeter of a certain sector of a circle of radius `6.5` cm is 31cm. Find the area of the sector.

Explanation

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

### Step 1: Understand the given information
- Radius (r) = 6.5 cm
- Perimeter of the sector = 31 cm

### Step 2: Write the formula for the perimeter of a sector
The perimeter (P) of a sector of a circle can be calculated using the formula:
\[ P = 2r + \frac{\theta}{360} \times 2\pi r \]
Where:
- \( r \) is the radius,
- \( \theta \) is the angle of the sector in degrees.

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 31 = 2(6.5) + \frac{\theta}{360} \times 2\pi(6.5) \]

### Step 4: Simplify the equation
Calculating \( 2(6.5) \):
\[ 2(6.5) = 13 \]
So, the equation becomes:
\[ 31 = 13 + \frac{\theta}{360} \times 13\pi \]

### Step 5: Isolate the term with \( \theta \)
Subtract 13 from both sides:
\[ 31 - 13 = \frac{\theta}{360} \times 13\pi \]
\[ 18 = \frac{\theta}{360} \times 13\pi \]

### Step 6: Solve for \( \theta \)
Multiply both sides by \( 360 \):
\[ 18 \times 360 = \theta \times 13\pi \]
\[ 6480 = \theta \times 13\pi \]
Now, divide both sides by \( 13\pi \):
\[ \theta = \frac{6480}{13\pi} \]

### 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 value of \( \theta \) and \( r \):
\[ A = \frac{\frac{6480}{13\pi}}{360} \times \pi (6.5)^2 \]

### Step 8: Simplify the area expression
First, calculate \( (6.5)^2 \):
\[ (6.5)^2 = 42.25 \]
Now substitute this into the area formula:
\[ A = \frac{6480 \times 42.25}{13 \times 360} \]

### Step 9: Calculate the area
Now calculate:
\[ A = \frac{274680}{4680} \]
\[ A = 25.5 \, \text{cm}^2 \]

### Final Answer
The area of the sector is \( 25.5 \, \text{cm}^2 \).

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