The area of the parellelagramm is base times height. You know figured out the height. So the base is the radius of the circle. You know that.
What is the area of the circle wedge? Well the entire circle is 360 degrees so this is just 30 degrees. It's a specific proportion of the entire circle. So if you can find the area of the entire circle you can find the area of the circle wedge.
The area of the shaded area is the area of the paralelagram minus the area of the circle wedge.
The circle fits snuggly between one pair of parallels, but is two large for the other pair. It follows that the diameter of the circle is the larger of the two distances between parallels (aka. altidudes). In other words, the diameter of the circle is $4\,\text{cm}$ and the height of the shaded region is the difference $4\,\text{cm}-3\,\text{cm}=1\,\text{cm}$.
Can you take it from here?
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Hint: Join the center of the circle to the points A and B. You'll obtain a triangle. What type of triangle is it?