Find the area of the sector of a circle of radius 7cm, if the corresponding arc length 6.2 cm.

Answer

To find the area of the sector of a circle with a radius of 7 cm and an arc length of 6.2 cm, we can follow these steps:

### Step 1: Understand the relationship between arc length, radius, and angle
The formula for the arc length \( L \) of a sector is given by:
\[ L = \frac{2 \pi r \theta}{360} \]
where \( r \) is the radius, \( \theta \) is the angle in degrees, and \( L \) is the arc length.

### Step 2: Substitute the known values into the arc length formula
We know:
- \( L = 6.2 \, \text{cm} \)
- \( r = 7 \, \text{cm} \)

Substituting these values into the formula:
\[ 6.2 = \frac{2 \pi (7) \theta}{360} \]

### Step 3: Solve for \( \theta \)
To isolate \( \theta \), we rearrange the equation:
\[ \theta = \frac{6.2 \times 360}{2 \pi (7)} \]

### Step 4: Calculate \( \theta \)
Now, we can calculate \( \theta \):
\[ \theta = \frac{6.2 \times 360}{2 \times 3.14 \times 7} \]
Calculating the numerator:
\[ 6.2 \times 360 = 2232 \]
Calculating the denominator:
\[ 2 \times 3.14 \times 7 = 43.96 \]
Now, divide:
\[ \theta = \frac{2232}{43.96} \approx 50.7 \, \text{degrees} \]

### Step 5: Use the angle to find the area of the sector
The area \( A \) of the sector can be calculated using the formula:
\[ A = \frac{\pi r^2 \theta}{360} \]
Substituting the known values:
\[ A = \frac{\pi (7^2) (50.7)}{360} \]

### Step 6: Calculate the area
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Substituting this value into the area formula:
\[ A = \frac{\pi (49) (50.7)}{360} \]
Calculating the numerator:
\[ 49 \times 50.7 = 2484.3 \]
Now, substituting this into the area formula:
\[ A = \frac{\pi (2484.3)}{360} \]
Using \( \pi \approx 3.14 \):
\[ A \approx \frac{3.14 \times 2484.3}{360} \approx \frac{7787.542}{360} \approx 21.6 \, \text{cm}^2 \]

### Final Answer
The area of the sector is approximately:
\[ \boxed{21.6 \, \text{cm}^2} \]