What is the area of a square inscribed in a semicircle problem going viral on social media

$s^2 = \frac{4}{5}R^2$ is correct for the standard orientation. For a **rotated square** inscribed in a semicircle, the problem changes significantly. A square of side length $s$ rotated by angle $\theta$ inscribed in a semicircle of radius $R$ would require solving: Two corners on the diameter: $(x_1, 0)$ and $(x_2, 0)$ with $|x_1 - x_2| = s\cos\theta$ Top corner on the arc: $(\frac{x_1+x_2}{2}, s\sin\theta)$ satisfying $x^2 + y^2 = R^2$ This is much more complex and the area would depend on $\theta$, with maximum at $\theta = 0$. **A more interesting variant:** A square inscribed in a QUARTER-circle (both sides touching the axes). Put the square in the corner with sides on the axes and the opposite corner on the quarter-circle arc. Then the diagonal of the square equals the radius, so $s\sqrt{2} = R$, giving area $s^2 = R^2/2$. This is a common Olympiad problem.