Notes
rectangle tilted over a square solution

Solution to the Rectangle Tilted Over a Square Puzzle

Solution by Similar Triangles

In the above diagram, triangles $A B C$ and $C D E$ are similar since they are right-angled triangles and angles $A \hat{C} B$ and $D \hat{C} E$ add together to give $90^\circ$. Therefore the ratio of lengths $A C : B C$ and $C E : D E$ are equal. Since $E D$ is twice $B C$, $E C$ is therefore twice $C A$, and hence is of length $8$. The shaded area is therefore $4 \times 8 = 32$.

Solution by Invariance Principle

The shaded rectangle can be tilted by varying amounts. Drawing it vertically, so that points $A$ and $B$ coincide, means that the outer square has side length $8$ and the height of the rectangle is the same as the side length.