What is the largest circle (or half circle) that can be inscribed in an arbitrary rectangle if a chord can be cut off and redistributed?
My mother, who is cutting material for sewing a skirt, posed the following problem. Suppose you take a bolt of cloth and fold it in half. She wants to know, what's a formula for the largest circle you can cut out if you redistribute the overflow to the opposite corner? (See drawing, below.)
My intuition was to first question how this gains anything over inscribing the circle without cutting any off? I realized it may be important to remember that if the fold is on the top edge of the figures in the drawing (long side of "A"), then the section "B" which is cut off will actually result in two pieces when its pattern is relocated to the bottom-left corner (because it's no longer on the fold).
It also occurs to me that whether there's a closed-form formula or a method of construction are two separate questions. It reminds me of the Moving Sofa Problem (but hopefully not as difficult).
Let your rectangle be $1$ unit high and call its width $d$. Two circles with radius $r$ are centered around $(r,0)$ and $(0,1+d-r)$ respectively. If they are tangent to each other the distance between their centers is $2r$
$$\sqrt{r^2+(1+d-r)^2}=2r \qquad r = \frac{\sqrt3 - 1}2 (d+1)$$
Note that $d \leq \sqrt3$ because $r \leq 1$. You can try some values with this desmos chart.