Step 1: Convert the angle to radians instead of using the degree-fraction formula.
The arc length formula in radian measure is
\[ l = r\theta \]
where $\theta$ is the central angle measured in radians. We first convert $60^{\circ}$ to radians using $180^{\circ} = \pi$ radians.
Step 2: Carry out the conversion.
\[ \theta = 60^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{\pi}{3} \text{ radians} \]
Step 3: Substitute into the radian arc-length formula.
With $r = 21\text{ cm}$ and $\pi \approx \frac{22}{7}$:
\[ l = r\theta = 21 \times \frac{\pi}{3} = \frac{21\pi}{3} = 7\pi \]
Step 4: Evaluate using $\pi \approx \frac{22}{7}$.
\[ l = 7 \times \frac{22}{7} = 22\text{ cm} \]
The 7 in the numerator cancels neatly with the 7 in the denominator of $\frac{22}{7}$.
Final Answer:
The length of the arc is $22\text{ cm}$, which corresponds to option (A).
\[ \boxed{22\text{ cm}} \]