A rectangle is inscribed in a semicircle of radius $R$ with base on the diameter. The **maximum area** is:

Half-width = $x$, height $y = \sqrt{R^2 - x^2}$. Area $A = 2xy = 2x\sqrt{R^2 - x^2}$. $dA/dx = 2\sqrt{R^2 - x^2} - 2x^2/\sqrt{R^2 - x^2} = 0$ ⇒ $R^2 = 2x^2$ ⇒ $x = R/\sqrt 2$. Then $y = R/\sqrt 2$, area = $2 \cdot (R/\sqrt 2)(R/\sqrt 2) = R^2$.