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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!

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    $\begingroup$ you mean radius 1? As the max area is $\pi$, the radius is 1. $\endgroup$
    – Notwen
    Commented Feb 6 at 10:32
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    $\begingroup$ Yes, it is the same as saying radius 1. $\endgroup$
    – A Math guy
    Commented Feb 6 at 10:33
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    $\begingroup$ I think you should also specify that the rectangles are not overlapping. $\endgroup$
    – Notwen
    Commented Feb 6 at 10:54
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    $\begingroup$ Apparently, the answer for three squares in a unit circle is unknown (or at any rate not proved): erich-friedman.github.io/packing/squincir $\endgroup$ Commented Feb 6 at 11:19
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    $\begingroup$ @Notwen They can overlap, so I didn't state it out. $\endgroup$
    – A Math guy
    Commented Feb 6 at 11:48

1 Answer 1

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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: $$A = 3x\sqrt{4-x^{2}}-\frac{3\sqrt{3}}{2}x^2$$ $$\frac{dA}{dx}=0\Rightarrow x=\sqrt{2-2\sqrt{\frac{3}{7}}}\approx 0.8311\Rightarrow \boxed{A=3\sqrt{7}-3\sqrt{3}}\approx 2.7411$$

enter image description here

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  • $\begingroup$ Can you proof that this is the largest among them all? Thanks. $\endgroup$
    – A Math guy
    Commented Feb 7 at 2:57

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