What is the maximum fractional coverage of the area of a circle by three, non-overlapping rectangles, all of which lie internal to the circle? (The rectangles can be of different sizes)
I started with the same assumption as Steven with the one large rectangle centered and two congruent small rectangles. While drawing a diagram I connected the two small rectangles into a large rectangle. Then I would only need to calculate two centered, orthogonal rectangles minus their overlap.
Then another idea occurred (and another assumption). I expect the final solution to be highly symmetric, then the two rectangles will be the same size, reducing the problem to a single variable.
WLOG, let the circle have radius 1. Let the short side of the rectangles be x, which makes the long side equal sqrt(4-x^2). Then the area covered equals A(x) = 2*x*sqrt(4-x^2) - x^2. This formula is valid as long as x <= sqrt(4-x^2), which implies x <= sqrt(2) = 1.4142
The local maximum of A(x) occurs at x=sqrt(2-2/sqrt(5))=1.0515 with A=2*(sqrt(5)-1)=2.472. Then the fraction of the circle covered equals 2*(sqrt(5)-1)/pi = 0.7869.
Possible solution
