Suppose we have a circle with a radius $R$ having a thread of length $L_0$ tied at diametrically opposite ends such that $2R < L_0 < \pi R$. What shape the thread must be in order to minimize the shaded area.
2026-04-23 02:00:43.1776909643
How to maximize area bounded by thread and a fixed line?
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You are asking the same as maximizing the surface area between the thread and the thread and the diameter, imposing that the thread stays below the half circle.
The solution to this problem is the arc circle: indeed we know that the circle has optimal area/perimeter ratio. Consider the circle of radius R' such that the circle arc between two points $A$, $B$ such that $AB = 2R$ is precisely $L_0$.
Assume that the arc circle is not the shape with optimal area/perimeter ratio, and call $\ell$ a better shape. Then the shape of the circle minus the arc, glued to $\ell$ would have the same perimeter as the circle, but a higher surface area, which implies a contradiction.
Thus the circle arc is optimal without the constraint of being below the half-circle. Since it is under the half-circle, then it is the optimal solution in this setting as well.