Olympiad Geometry · Superchallenge problem

A cone-shaped glass is held with its point straight down. Its open top is a circle of radius $12$ cm and it is $16$ cm deep. It contains some water. A solid ball of radius $6$ cm is lowered in until it rests against the glass all the way around. Now the water level is exactly at the rim, just covering the top of the ball, and no water has spilled. How deep was the water before the ball went in? Give your answer in simplest radical form.