Notes
three overlapping circles solution

Solution to the Three Overlapping Circles Puzzle

Solution by Dissection and Circle Area

It was noted in the twitter thread that the centres of each circle lie at the intersections of the other two. This means that the radii of the circles is $4$.

Consider the blue region with vertices $A$, $B$, and $C$. The curved area below the line segment $A B$ is congruent to the purple shaded region, meaning that the blue region has the same area as the region outlined in green. This is a sector with central angle $60^\circ$ and radius $4$. Taking three of them gives a semi-circle so has area $\frac{1}{2} \pi 4^2 = 8 \pi$.