Question

The length of the arc of the sector of a circle with radius $21\text{ cm}$ and of central angle $60^\circ$, is :

• A. $22\text{ cm}$
• B. $44\text{ cm}$
• C. $88\text{ cm}$
• D. $11\text{ cm}$

Hint:

An angle of $60^\circ$ is exactly $\frac{1}{6}$ of a full rotation ($360^\circ$).
Therefore, the arc length is simply one-sixth of the total circumference of the circle:
\[ \text{Circumference} = 2 \times \frac{22}{7} \times 21 = 132\text{ cm} \]
\[ \text{Arc Length} = \frac{132}{6} = 22\text{ cm} \]
Recognizing common fractional parts of a circle (like $\frac{1}{6}$ for $60^\circ$ or $\frac{1}{4}$ for $90^\circ$) helps solve these problems rapidly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic is Areas Related to Circles, specifically finding the length of an arc of a circle.
We are given a circle of radius $21\text{ cm}$ and a sector within it that subtends an angle of $60^\circ$ at the centre of the circle.
We need to determine the length of the corresponding arc of this sector.

Step 2: Key Formula or Approach:
The length of an arc ($l$) of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is given by the formula:
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \]
Here, we will use the standard approximation of $\pi = \frac{22}{7}$ for calculations.

Step 3: Detailed Explanation:

• Identify the given parameters from the problem:
Radius of the circle, $r = 21\text{ cm}$
Central angle of the sector, $\theta = 60^\circ$

• Write down the formula for the arc length:
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \]

• Substitute the given values into this expression:
\[ l = \frac{60^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 21 \]

• Simplify the fraction representing the fractional part of the circle:
\[ \frac{60^\circ}{360^\circ} = \frac{1}{6} \]

• Now substitute this simplified fraction back into the equation:
\[ l = \frac{1}{6} \times 2 \times \frac{22}{7} \times 21 \]

• Perform the division of 21 by 7:
\[ \frac{21}{7} = 3 \]
This simplifies the product to:
\[ l = \frac{1}{6} \times 2 \times 22 \times 3 \]

• Combine the terms in the numerator:
\[ 2 \times 3 = 6 \]
Now, substitute this back:
\[ l = \frac{1}{6} \times 6 \times 22 \]

• Cancel the 6 in the numerator and denominator:
\[ l = 22\text{ cm} \]


Step 4: Final Answer:
The length of the arc of the sector is $22\text{ cm}$, which corresponds to option (A).