Find the area of the shaded region
the radius of each circle is 1
1. Three of the four midpoints form a right triangle with the length of the hypotenuse of 2. Thus the distance between twoo midpoints placed on a vertical (or horizontal) line is
$\displaystyle d = \sqrt{2}$
2. The triangle formed by a midpoint and two points of interception (marked blue) is a right triangle whose area is
$\displaystyle a = \frac12 \cdot r^2$
3. The lens shaped area is the difference of a quarter circle and the blue right triangle
$\displaystyle l = \frac14 \cdot \pi r^2-\frac12 \cdot r^2 = \frac14 r^2(\pi - 2)$
4. The shaded region's area is the difference of 4 circles and 8 lens shaped areas.