Wicked circle

Let $O$ be the centre of the circle and $r$ be its radius. Let the perpendicular from $O$ to side $ST$ intersect this side at $P$ . Then $\Delta SOP$ is right-angled at $P$ with $|OP| = r, |SP| = 2 - r$ and $\angle OSP = 30^{\circ}$ . Thus

$\tan(\angle OSP) = \dfrac{|OP|}{|SP|} \Longrightarrow \tan(30^{\circ}) = \dfrac{r}{2 - r} \Longrightarrow \dfrac{1}{\sqrt{3}} = \dfrac{r}{2 - r} \Longrightarrow 2 - r = \sqrt{3}r \Longrightarrow 2 = (1 + \sqrt{3})r \Longrightarrow r = \dfrac{2}{1 + \sqrt{3}} = \sqrt{3} - 1 \approx \boxed{0.732}$ .