Notes
triangle in semi-circle with square solution

Solution to the Triangle in Semi-Circle with Square Puzzle

Solution by Lengths in an Equilateral Triangle and Pythagoras' Theorem

With the points labelled as above, the radius of the semi-circle is $3$ and this is the height of the equilateral triangle. From lengths in an equilateral triangle, the length of $O A$ is $\sqrt{3}$. Applying Pythagoras' theorem to triangle $O A B$ shows that the square of the side length of the square is $3^2 - 3 = 6$ and so the area of the square is $6$.