What is the maximum area that can be covered with $3$ rectangles inside a radius $1$ circle?(i.e. maximum area $=\pi$) The rectangles can be any length and height you want, and can rotate and reflect.
(I found a solution which is $1.5\sqrt{3}$, which is to make a **hexagon shape **outline)
Bonus: Proof why this shape you found is the largest possible, and is the answer related to those questions which allows more rectangles in the circle(e.g. you can put $4$ inside.)
Bonus 2: Does your method also work on ANY four-sided shapes(like a kite)? Thanks!
By using a hexagonal pattern but offsetting the corners, we can slightly increase the area. If the short sides of the rectangles are all $x<1$, we have: $$A = 3x\sqrt{4-x^{2}}-\frac{3\sqrt{3}}{2}x^2$$ $$\frac{dA}{dx}=0\Rightarrow x=\sqrt{2-2\sqrt{\frac{3}{7}}}\approx 0.8311\Rightarrow \boxed{A=3\sqrt{7}-3\sqrt{3}}\approx 2.7411$$