Largest rectangle inscribed in a circle

Text

Find the maximum area of a rectangle inscribed in a circle of radius $R=5$.

Solution

Let the sides be $2x$ and $2y$ with $x^{2}+y^{2}=R^{2}$; area $A=4x\sqrt{R^{2}-x^{2}}$. Maximizing $A^{2}=16x^{2}(R^{2}-x^{2})$ gives $R^{2}-2x^{2}=0\Rightarrow x=\dfrac{R}{\sqrt2}=y$ (a square). Side $=R\sqrt2=5\sqrt2\approx7.07$; area $A=2R^{2}=50$. $\boxed{A_{\max}=2R^{2}=50\ \text{(square of side }5\sqrt2\approx7.07)}$