First, find the length of BC using the tangent of angle 60 degrees (\(\frac{\pi}{3}\) radians):

\(\tan 60 = \frac{BC}{6}\)

\(BC = 6\sqrt{3}\)

Next, calculate the area of triangle OBC:

\(\text{Area of } \triangle OBC = \frac{1}{2} \times 6 \times 6\sqrt{3} = 18\sqrt{3}\)

Calculate the area of sector AOC:

\(\text{Area of sector } AOC = \frac{1}{2} \times 6^2 \times \frac{\pi}{3} = 6\pi\)

Finally, find the area of the shaded region by subtracting the area of the sector from the area of the triangle:

\(\text{Area of shaded region} = 18\sqrt{3} - 6\pi\)