Intersection area of multiple circles

Question

Every other point of a regular hexagon with a side length a is the center of a circle whose radius is equal to the shortest diagonal of that hexagon. Calculate the area of intersection of these circles. The diagram is here (The asked area is the ACE intersection): https://i.stack.imgur.com/bY9Fj.jpg

Answer 1

The region is obtained from three 60°-sectors of a disc, which all overlap in an equilateral triangle. Hence the total area is that of a half disk minus two triangles. You can find the radius of the disc/side of the triangle with Pythagoras.

Answer 2

Length of the diagonal is $2 a \cos \frac{\pi}{6} = \sqrt 3 a$. With that and the answer from Hagen von Eitzen, you should get the result.