Radius of a sector of a circle is 3.5 cm and length of its arc is 2.2 cm. Find the area of the sector.

Explanation

To find the area of the sector of a circle given the radius and the length of the arc, we can follow these steps:

### Step 1: Identify the given values
- Radius (r) = 3.5 cm
- Length of the arc (L) = 2.2 cm

### Step 2: Use the formula for the length of the arc
The formula for the length of the arc (L) is given by:
\[ L = \theta \cdot r \]
where \( \theta \) is the angle in radians.

### Step 3: Rearrange the formula to find \( \theta \)
From the formula, we can express \( \theta \) as:
\[ \theta = \frac{L}{r} \]
Substituting the values:
\[ \theta = \frac{2.2 \, \text{cm}}{3.5 \, \text{cm}} \]

### Step 4: Calculate \( \theta \)
Calculating \( \theta \):
\[ \theta = \frac{2.2}{3.5} = \frac{22}{35} \, \text{radians} \]

### Step 5: Use the formula for the area of the sector
The area (A) of the sector is given by:
\[ A = \frac{1}{2} \theta r^2 \]
Substituting the values of \( \theta \) and \( r \):
\[ A = \frac{1}{2} \cdot \frac{22}{35} \cdot (3.5)^2 \]

### Step 6: Calculate \( (3.5)^2 \)
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]

### Step 7: Substitute back into the area formula
Now substituting back into the area formula:
\[ A = \frac{1}{2} \cdot \frac{22}{35} \cdot 12.25 \]

### Step 8: Calculate the area
Calculating the area:
\[ A = \frac{22 \cdot 12.25}{2 \cdot 35} \]
\[ A = \frac{270.5}{70} \]
\[ A = 3.85 \, \text{cm}^2 \]

### Final Answer:
The area of the sector is \( 3.85 \, \text{cm}^2 \).

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