Determine the dimensions of the cylinder and the height of the unsubmerged part.

Radius $r = 6$ cm.
Length $L = 20$ cm.
Diameter $D = 2r = 2 \times 6 = 12$ cm.
The solid submerges three quarters of its diameter, so the unsubmerged part is one quarter of its diameter.
Height of the unsubmerged segment $h = \frac{1}{4} \times D = \frac{1}{4} \times 12 = 3$ cm.