Notes
rectangle overlapping a circle solution

Solution to the Rectangle Overlapping a Circle Puzzle

Solution by Angle in a Semi-Circle and Angles in the Same Segment

With the points labelled as in the diagram, angle $A \hat{D} B$ is a right-angle so using the result that the angle in a semi-circle is a right-angle, $A B$ is a diameter of the circle. Then by the same result, angle $A \hat{C} B$ is also a right-angle. This means that the marked angle is $30^\circ$.

Angle $D \hat{C} A$ is then $30^\circ$ and so as angles in the same segment are equal, angle $D \hat{B} A$ is also $30^\circ$. So triangle $A B D$ is a $30^\circ-60^\circ-90^\circ$ right-angled triangle, which means it is half an equilateral triangle. This means that the length of $B D$ is $\frac{\sqrt{3}}{2}$ times the length of $A B$, so $A B$ has length $\frac{2}{\sqrt{3}} \times 3 = 2 \sqrt{3}$ and the radius of the circle is therefore $\sqrt{3}$. Its area is then $3 \pi$.