Find the area of the shaded region, circle and parallelogram

Question

Given a circle with center A and radius 2. If ABCD is a parallelogram, find the area of the shaded region.

I am having trouble with this problem. I know that:

-This is a 30:60:90 triangle, so the height is 1, and the other side length is root 3. Also, segment CD is also 2, and angle BCD is 30° because it is a parallelogram

...and that's pretty much it. Could someone help explain how to solve the rest of the problem?

Answer 1

Hint:-

Area of region sector ABD=$\frac12r^2\theta$.

Now,subtract it from the total parallelogram area.

Proof for the inquisitive mind:- Area of full circle=$\pi r^2$

Full rotation measures $2\pi$ radians.

Now comparing the two results with a bit of unitary method we get area of sector=$\pi r^2*\frac{\theta}{2\pi}$.

https://proofwiki.org/wiki/Area_of_Sector

Answer 2

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.