Right triangle with uniform legs and a wide angle

Build a right triangle whose two legs $A$ and $B$ have lengths drawn independently from $\text{Unif}(0,1)$. Let $\theta$ denote the acute angle lying opposite leg $A$. The probability that $\theta > \pi/3$ can be written as $\dfrac{a}{b\sqrt{c}}$ with $a,b,c$ positive integers that are relatively prime. Compute $abc$.

$6$. Since $\tan\theta = A/B$, the event is $A>\sqrt3\,B$. In the unit square this is a triangle with legs $1$ and $1/\sqrt3$, of area $\tfrac{1}{2\sqrt3}$, so $a=1,b=2,c=3$ and $abc=6$.