What is the largest area of a rectangle that fits inside a circle of radius 5 cm, in cm²?

Explanation

1. 01

   The rectangle's diagonal is the diameter, 10. With sides $x$ and $y$, $x^2 + y^2 = 100$ and the area is $A = x\sqrt{100 - x^2}$.

2. 02

   $A^2 = 100x^2 - x^4$ is largest when $200x - 4x^3 = 0$, that is $x^2 = 50$: then $y^2 = 50$ too, a square, and $A = 50$.