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: Use the radian version of the arc length formula instead of the degree version.
Instead of writing the arc length as a fraction of the circumference, we can convert the angle to radians first and use the simpler formula $l = r\theta$.

Step 2: Convert the given angle to radians.
We know $180^\circ = \pi$ radians, so:
\[ 60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3}\text{ radians} \]

Step 3: Apply the radian arc length formula.
For a circle of radius $r$ and an angle $\theta$ measured in radians, the arc length is:
\[ l = r\theta \]
Substitute $r = 21\text{ cm}$ and $\theta = \frac{\pi}{3}$:
\[ l = 21 \times \frac{\pi}{3} = 7\pi \]

Step 4: Substitute the value of $\pi$ and simplify.
Using $\pi = \frac{22}{7}$:
\[ l = 7 \times \frac{22}{7} = 22\text{ cm} \]

Step 5: Final answer.
The arc length is $22\text{ cm}$, which is option (A).
\[ \boxed{l = 22\text{ cm}} \]