Notes
two quarter circles in a square solution

Solution to the Two Quarter Circles in a Square Puzzle

Solution by Area of a Circle and Area of a Triangle

As this is a question about what fraction of the square is shaded, let us set the side length of the square to $1$.

With the points labelled as above, the sector $A F B$ is an eighth of a circle with centre $B$ and radius $A B$, so its area is $\frac{\pi}{8}$.

As $F$ is on the quarter circle with centre $B$, the line segment $B F$ is the same length as $A B$. Triangle $B F E$ is an isosceles right-angled triangle and so the length of $B E$ is $\frac{1}{\sqrt{2}}$. The area of the quarter circle $B E F$ is therefore:

The area of triangle $B E F$ is then $\frac{1}{2} \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} = \frac{1}{4}$ so the segment cut out from the smaller quarter circle by the chord $B F$ has area $\frac{\pi}{8} - \frac{1}{4}$.

The shaded area is therefore:

And so the shaded area is one quarter of the square.