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

Answer

To solve the problem of finding the length of the arc and the area of a sector cut off from a circle with a radius of 21 cm and an angle of 150 degrees, we can follow these steps:

### Step 1: Identify the formulas
The formulas we need are:
- Length of the arc (L) of the sector:
\[ L = \frac{\theta}{360^\circ} \times 2\pi r \]
- Area (A) of the sector:
\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]

### Step 2: Substitute the known values
Given:
- Radius \( r = 21 \) cm
- Angle \( \theta = 150^\circ \)

Now we can substitute these values into the formulas.

### Step 3: Calculate the length of the arc
Using the formula for the length of the arc:
\[ L = \frac{150}{360} \times 2\pi \times 21 \]

### Step 4: Simplify the length calculation
First, simplify \( \frac{150}{360} \):
\[ \frac{150}{360} = \frac{5}{12} \]
Now substitute this back into the equation:
\[ L = \frac{5}{12} \times 2\pi \times 21 \]
\[ = \frac{5 \times 2 \times 22}{7 \times 12} \times 21 \]
\[ = \frac{10 \times 22 \times 21}{7 \times 12} \]

### Step 5: Perform the calculations
Calculating \( 10 \times 22 = 220 \) and \( 220 \times 21 = 4620 \):
\[ L = \frac{4620}{84} = 55 \text{ cm} \]

### Step 6: Calculate the area of the sector
Using the formula for the area of the sector:
\[ A = \frac{150}{360} \times \pi \times (21)^2 \]
\[ = \frac{5}{12} \times \pi \times 441 \]
Substituting \( \pi = \frac{22}{7} \):
\[ A = \frac{5}{12} \times \frac{22}{7} \times 441 \]

### Step 7: Simplify the area calculation
Calculating \( 441 \div 12 = 36.75 \):
\[ A = \frac{5 \times 22 \times 36.75}{7} \]
Calculating \( 5 \times 22 = 110 \):
\[ A = \frac{110 \times 36.75}{7} \]
Calculating \( 110 \times 36.75 = 4042.5 \):
\[ A = \frac{4042.5}{7} = 577.5 \text{ cm}^2 \]

### Final Answers
- Length of the arc: \( 55 \) cm
- Area of the sector: \( 577.5 \) cm²