Maximum area of a rectangle inscribed in a semicircle of radius R (base on diameter):

Width = 2x, height = sqrt(R² - x²). Area A = 2x sqrt(R² - x²). dA/dx = 0 gives x = R/sqrt(2). A_max = 2(R/sqrt(2)) sqrt(R² - R²/2) = sqrt(2) R × (R/sqrt(2)) × sqrt(2) = R² sqrt(2). Wait, recompute: A = 2 × (R/√2) × √(R²/2) = 2 × (R/√2) × (R/√2) = R². So max area = R². (Width = R√2, height = R/√2.)