Find the area of region.

Question

A chord of length R divides a circular area of radius R into two regions. Find the sides of the rectangle with the largest area that can be inscribed in the smaller region with one side along the given chord. Please provide me some hint in solving this problem.

Answer 1

Assume $R=1$ without loss of generality, use the trigonometric circle and let the chord be vertical $x=S$. Let one of the rectangle vertices be at $(\cos t, \sin t)$, with $\cos t\le S$. The rectangle area is given by $2(\cos t-S)\sin t$, which you need to maximize.