HMMT November 2014 U29

Let $\omega$ be a fixed circle with radius 1 , and let $B C$ be a fixed chord of $\omega$ such that $B C=1$. The locus of the incenter of $A B C$ as $A$ varies along the circumference of $\omega$ bounds a region $\mathcal{R}$ in the plane. Find the area of $\mathcal{R}$.