We want the football-shaped area in red below:


 

Draw a line segment connecting their centers, 
and line segments from the centers to their upper 
point of intersection:
 

 
That green triangle is an equilateral triangle because all its 
sides are radii of one circle or the other.
 
Therefore the angle marked is 60°, and all the green sides of
the equilateral triangle are "r" in length.
 
Now let's draw another equilateral triangle underneath it:


 
Those two 60° angles make a big 120° angle, so let's erase 
three of those green lines and mark that big angle 120°.



Now we'll lake away everything but the  (D-shaped) 
segment outlined in red, and leave the two green radii 
which make up the central angle of 120° which subtends 
the (D-shaped) segment outlined in red.

 

The formula for the area of a (D-shaped) sector of a circle is



where the subtending central angle is measured in radians.

We convert 120° to radians:   and substitute in the formula:


 



That's the area of this (D-shaped) segment, which is half of the 
desired area:



So the desired area of the entire overlapping part




is twice the area of the (D-shaped) sector or



or



Edwin