# Find the area of the shaded region, circle and parallelogram

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?

• Find the area of the parellelagram, the area of the circle arc. And subtract. – fleablood Oct 14 '15 at 2:52
• Can you compute the area of the parallelogram ? Then try to compute the area of the section of disc $ABD$. – Joel Cohen Oct 14 '15 at 2:53

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.

• How would one find the area of the circle wedge? – Dana Oct 14 '15 at 3:03
• 30 degrees is 1/12 of 360 degrees so the area of the wedge will be 1/12 of the area of the circle. – fleablood Oct 14 '15 at 3:10
• Okay. I think I'm understanding it now. So I find the area of the circle, take 1/12 of that, and subtract that value from the area of the parallelogram? – Dana Oct 14 '15 at 3:13

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