Area of the part of $120^{\circ }$ of a circle is $\frac{66}{7}{\text{ cm}}^2$ then diameter of the circle is ($\pi =22/7$):

The area of a sector of a circle with angle $\theta$ is given by:
$$\text{Area}=\frac{\theta }{360^{\circ }}\times \pi r^2$$
Substitute the given parameters:
$$\frac{66}{7}=\frac{120^{\circ }}{360^{\circ }}\times \frac{22}{7}\times r^2$$
$$\frac{66}{7}=\frac{1}{3}\times \frac{22}{7}\times r^2$$
Cancel out 7 from denominators on both sides:
$$66=\frac{22}{3}\times r^2$$
$$3=\frac{1}{3}\times r^2⟹r^2=9⟹r=3\text{ cm}$$
$$\text{Diameter}=2r=2\times 3=6\text{ cm}$$