Find the area of the sector of a circle of radius 5 cm, if the corresponding arc length is 3.5 cm.

Answer

To find the area of the sector of a circle with a radius of 5 cm and an arc length of 3.5 cm, we can follow these steps:

### Step 1: Understand the formulas
We need to use two formulas:
1. The formula for the length of the arc (L) of a sector:
\[ L = \frac{\theta}{360} \times 2\pi r \]
where \( \theta \) is the central angle in degrees, and \( r \) is the radius.

2. The formula for the area (A) of the sector:
\[ A = \frac{\theta}{360} \times \pi r^2 \]

### Step 2: Substitute the known values into the arc length formula
Given:
- Radius \( r = 5 \) cm
- Arc length \( L = 3.5 \) cm

We can rearrange the arc length formula to find \( \theta \):
\[ 3.5 = \frac{\theta}{360} \times 2\pi \times 5 \]

### Step 3: Solve for \( \theta \)
Rearranging the equation:
\[ 3.5 = \frac{\theta}{360} \times 10\pi \]
\[ \theta = \frac{3.5 \times 360}{10\pi} \]
Calculating \( \theta \):
\[ \theta = \frac{1260}{10\pi} = \frac{126}{\pi} \text{ degrees} \]

### Step 4: Substitute \( \theta \) into the area formula
Now we can find the area of the sector using the area formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
Substituting \( \theta \) and \( r \):
\[ A = \frac{\frac{126}{\pi}}{360} \times \pi \times (5^2) \]
\[ A = \frac{126}{360} \times 25 \]
Calculating \( A \):
\[ A = \frac{126 \times 25}{360} \]
\[ A = \frac{3150}{360} = 8.75 \text{ cm}^2 \]

### Final Answer
The area of the sector is **8.75 cm²**.

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