Right triangle with uniform legs: angle above 45 degrees

A right triangle is built whose two legs, $A$ and $B$, have independent lengths drawn uniformly from $(0,1)$. Let $\theta$ be the acute angle facing leg $A$. What is the probability that $\theta$ exceeds $\pi/4$?

$\tfrac12$. Since $\tan\theta=A/B$, the condition $\theta>\pi/4$ is the same as $A>B$. The two legs are i.i.d., so by symmetry this happens with probability $\tfrac12$.