# Covering a circle using rectangles

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!

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

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