For a square inscribed in a semicircle of radius r = 12 cm, one side of the square lies along the diameter of the semicircle, and the opposite two vertices touch the semicircular arc. Let the square have side length s. If we draw the semicircle with the flat side (diameter) horizontal and the square sitting on it, the top vertices of the square lie on the semicircle. The center of the semicircle is at the midpoint of the diameter. For the square to be maximized, its top corners lie exactly on the semicircle. Using the property that for a square inscribed in a semicircle, the diagonal from the center to a top vertex forms a right triangle: (s/2)² + s² = r², giving s²/4 + s² = 144, or 5s²/4 = 144, so s² = 576/5 = 115.2. The area of the square is s² = 115.2 cm².