Olympiad Geometry · Superchallenge problem

A hemispherical bowl of radius $6$ is filled to the brim with water. It is then tipped so that the flat circle of its rim makes a $30^\circ$ angle with the horizontal. How much water spills out? Leave your answer in terms of $\pi$. (A cap of height $h$ cut from a ball of radius $R$ has volume $\tfrac13\pi h^2(3R - h)$.)