A sector is cut from a circle of radius 21 cm. The angle of the sector is `150^(@)`. Find its area

Answer

To find the area of a sector of a circle, we can use the formula:

\[ \text{Area of Sector} = \frac{\theta}{360} \times \pi r^2 \]

where:
- \(\theta\) is the angle of the sector in degrees,
- \(r\) is the radius of the circle,
- \(\pi\) is a constant approximately equal to \(3.14\) or can be taken as \(\frac{22}{7}\) for calculations.

Given:
- Radius \(r = 21 \, \text{cm}\)
- Angle \(\theta = 150^\circ\)

### Step 1: Substitute the values into the formula

\[ \text{Area of Sector} = \frac{150}{360} \times \frac{22}{7} \times (21)^2 \]

### Step 2: Calculate \( (21)^2 \)

\[ (21)^2 = 441 \]

### Step 3: Substitute \( (21)^2 \) back into the area formula

\[ \text{Area of Sector} = \frac{150}{360} \times \frac{22}{7} \times 441 \]

### Step 4: Simplify \(\frac{150}{360}\)

\[ \frac{150}{360} = \frac{5}{12} \]

### Step 5: Substitute the simplified fraction back into the area formula

\[ \text{Area of Sector} = \frac{5}{12} \times \frac{22}{7} \times 441 \]

### Step 6: Calculate \(\frac{22}{7} \times 441\)

\[ \frac{22}{7} \times 441 = \frac{22 \times 441}{7} = 22 \times 63 = 1386 \]

### Step 7: Substitute this value back into the area formula

\[ \text{Area of Sector} = \frac{5}{12} \times 1386 \]

### Step 8: Calculate \(\frac{5 \times 1386}{12}\)

\[ 5 \times 1386 = 6930 \]

Now divide by 12:

\[ \frac{6930}{12} = 577.5 \]

### Final Answer

The area of the sector is:

\[ \text{Area of Sector} = 577.5 \, \text{cm}^2 \]