An adventitious quadrangle is a (convex) quadrilateral with the property that if its two diagonals are drawn in, all angles formed are rational multiples of $\pi$. A classification of all such quadrilaterals was published by Bol, whose method was essentially correct but whose list contained some errors. A correct classification was published by Poonen and Rubinstein. It turns out that there are 65 "sporadic" solutions. The rational numbers arising in these sporadic solutions could be considered exotic, although the largest denominator that arises is "only" 210. One such sporadic adventitious quadrilateral has interior angles (in cyclic order) $139\pi/210$, $5\pi/14$, $6\pi/7$, and $13\pi/105$.

Note that adventitious angles have shown up in the popular press occasionally, e.g., the Washington Post in 1995 and in a United Airlines magazine in 2004. It's typically phrased as an innocent-looking "find the angle in the diagram" problem that looks easy but is actually very difficult, at least if trigonometry is not allowed.