In a circle of radius 21 cm, an arc subtends an angle of `60^(@)` at the centre. Find the area of sector formed by the arc.

Answer

To find the area of the sector formed by the arc in a circle of radius 21 cm that subtends an angle of 60 degrees at the center, we can follow these steps:

### Step 1: Identify the formula for the area of a sector
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360} \times \pi r^2 \]
where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle.

### Step 2: Substitute the known values into the formula
Here, the radius \( r = 21 \) cm and the angle \( \theta = 60 \) degrees. We also use \( \pi \approx \frac{22}{7} \) for our calculations. Substituting these values into the formula gives:
\[ A = \frac{60}{360} \times \frac{22}{7} \times (21)^2 \]

### Step 3: Simplify the fraction
First, simplify \( \frac{60}{360} \):
\[ \frac{60}{360} = \frac{1}{6} \]
Now substitute this back into the area formula:
\[ A = \frac{1}{6} \times \frac{22}{7} \times (21)^2 \]

### Step 4: Calculate \( (21)^2 \)
Calculate \( (21)^2 \):
\[ (21)^2 = 441 \]
Now substitute this value into the area formula:
\[ A = \frac{1}{6} \times \frac{22}{7} \times 441 \]

### Step 5: Multiply the fractions
Now, multiply the fractions:
\[ A = \frac{22 \times 441}{6 \times 7} \]

### Step 6: Calculate \( 6 \times 7 \)
Calculate \( 6 \times 7 \):
\[ 6 \times 7 = 42 \]
So now we have:
\[ A = \frac{22 \times 441}{42} \]

### Step 7: Simplify \( \frac{22 \times 441}{42} \)
Now, calculate \( 22 \times 441 \):
\[ 22 \times 441 = 9702 \]
Now divide by 42:
\[ A = \frac{9702}{42} \]

### Step 8: Perform the division
Now perform the division:
\[ A = 231 \]

### Final Answer
Thus, the area of the sector formed by the arc is:
\[ \text{Area} = 231 \text{ cm}^2 \]

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