Notes
a semi-circle and quarter circle in a square solution

Solution to the A Semi-Circle and Quarter Circle in a Square Puzzle

Solution by Area of a Circle and Lengths in a Square

Using the area of a circle, the radius of the quarter circle is $2$ and of the semi-circle is $\sqrt{2}$. This means that the point labelled $B$ in the above is the centre of a square with side length $2\sqrt{2}$ and diagonal $4$. Its area is therefore $8$.