The radius of a circle is 14 cm and the area of the sector is `102.7 cm^(2)`. Find the central angle of the sector.

Answer

To find the central angle of the sector given the radius of the circle and the area of the sector, we can follow these steps:

### Step 1: Write down the formula for the area of a sector
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where:
- \( A \) is the area of the sector,
- \( \theta \) is the central angle in degrees,
- \( r \) is the radius of the circle.

### Step 2: Substitute the known values into the formula
We are given:
- Radius \( r = 14 \, \text{cm} \)
- Area of the sector \( A = 102.7 \, \text{cm}^2 \)

Substituting these values into the formula:
\[ 102.7 = \frac{\theta}{360} \times \pi \times (14)^2 \]

### Step 3: Simplify the equation
First, calculate \( \pi \times (14)^2 \):
\[ \pi \times (14)^2 = \pi \times 196 = 196\pi \]
Thus, the equation becomes:
\[ 102.7 = \frac{\theta}{360} \times 196\pi \]

### Step 4: Solve for \( \theta \)
To isolate \( \theta \), multiply both sides by \( 360 \):
\[ 102.7 \times 360 = \theta \times 196\pi \]
Now, divide both sides by \( 196\pi \):
\[ \theta = \frac{102.7 \times 360}{196\pi} \]

### Step 5: Calculate the value of \( \theta \)
Using \( \pi \approx \frac{22}{7} \):
\[ \theta = \frac{102.7 \times 360}{196 \times \frac{22}{7}} \]
Calculating the denominator:
\[ 196 \times \frac{22}{7} = \frac{4312}{7} \]
Now substituting back:
\[ \theta = \frac{102.7 \times 360 \times 7}{4312} \]

### Step 6: Perform the calculations
Calculating \( 102.7 \times 360 \times 7 \):
\[ 102.7 \times 360 = 36972 \]
Then,
\[ 36972 \times 7 = 258804 \]
Now divide by \( 4312 \):
\[ \theta = \frac{258804}{4312} \approx 60.019 \]

### Step 7: Round the answer
Thus, rounding to the nearest whole number, we find:
\[ \theta \approx 60^\circ \]

### Final Answer
The central angle of the sector is \( 60^\circ \).

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