Proof Problem

It would help if they had drawn the figure so that it fit the description (two arcs rather than an ellipse).

Here is an adjusted copy of your image:
FMH115460 problem.png

Here is what it should look like:
FMH115460.png
See if that helps. Think about equilateral triangles and sectors of circles.

Be sure to show us what you have tried, so we can guide your thinking.
 
Well I have determined the following:
Area of a triangle = (2sqr3 x 2 ) /2 = 2* sqrt3
Area of sector = 90/360 * pi x (2 sqrt 3) ^2
1/4 pi x 12 = 3 pi

Area of shaded region = 3 pi = 2 sqrt 3

* 4 as four regions

= 4(3pi - 2 sqrt 3)

So as you can see I don't arrive at the answer
 
Check the area of the triangle. You're off by a factor of 2.

Check the area of the sector. The angle is not 90 degrees, and the radius is not 2 sqrt(3)!
 
Ok so I get that the area of each equilateral triangle is 4 root 3, so would the radius be 2?
 
Yes. No. Continue ...

I'm assuming you didn't really mean 2. The radius is easy to see in the picture.
 
You have looked at my version of the picture, which is not an ellipse, right??

If an arc has its center at A, and passes through B, C, and D, what is its radius? Imagine putting the point of the compass at A, ...
 
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