Step-by-step Solution:

To find the area of the sector, we use the formula \(A = \frac{\theta}{360^\circ} \times \pi r^2\).

1. Identify the given values:
Radius \(r = 6\) cm.
Angle of the sector \(\theta = 60^\circ\).

2. Substitute the values into the formula:
The options are in fractional form, so we will use \(\pi = \frac{22}{7}\).
\[\begin{aligned} A_{\text{sector}} &= \frac{60^\circ}{360^\circ} \times \pi (6)^2 \\ &= \frac{1}{6} \times \pi \times 36 \\ &= 6\pi \end{aligned}\]
3. Calculate the final area:
Now, substitute the value of \(\pi\).
\[\begin{aligned} A_{\text{sector}} &= 6 \times \frac{22}{7} \\ &= \frac{132}{7} \end{aligned}\]
4. Convert to a mixed fraction:
To convert \(\frac{132}{7}\) to a mixed fraction, we divide 132 by 7.
The quotient is 18 and the remainder is 6.
\[\frac{132}{7} = 18\frac{6}{7} \text{ cm}^2\]
Result:
The area of the sector is \(18\frac{6}{7} \text{ cm}^2\).