Find the area of sector of a circle of radius 5 m bounded by an arc of length 8 m.

Answer

To find the area of the sector of a circle with a radius of 5 m and an arc length of 8 m, we can follow these steps:

### Step 1: Identify the given values
- Radius (R) = 5 m
- Arc length (L) = 8 m

### Step 2: Calculate the central angle (θ) in radians
The formula to find the central angle in radians is:
\[ \theta = \frac{L}{R} \]
Substituting the values:
\[ \theta = \frac{8}{5} \text{ radians} \]

### Step 3: Convert the central angle from radians to degrees
To convert radians to degrees, we use the formula:
\[ \theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} \]
Substituting the value of θ:
\[ \theta_{degrees} = \frac{8}{5} \times \frac{180}{\pi} \]
Calculating this gives:
\[ \theta_{degrees} = \frac{1440}{5\pi} \approx 288^\circ \]

### Step 4: Use the formula for the area of the sector
The formula for the area of a sector is:
\[ \text{Area} = \frac{\theta}{360} \times \pi R^2 \]
Substituting the values:
\[ \text{Area} = \frac{288}{360} \times \pi \times (5^2) \]
Calculating \(5^2\):
\[ 5^2 = 25 \]
Now substituting back:
\[ \text{Area} = \frac{288}{360} \times \pi \times 25 \]

### Step 5: Simplify the expression
First, simplify \(\frac{288}{360}\):
\[ \frac{288}{360} = \frac{4}{5} \]
Now substituting this back:
\[ \text{Area} = \frac{4}{5} \times \pi \times 25 \]
Calculating this gives:
\[ \text{Area} = \frac{100\pi}{5} = 20\pi \text{ m}^2 \]

### Step 6: Final answer
Therefore, the area of the sector is:
\[ \text{Area} \approx 20 \text{ m}^2 \]