A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the other rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring? $\mathrm{(A)}\ 171\qquad$ $\mathrm{(B)}\ 173\qquad$ $\mathrm{(C)}\ 182\qquad$ $\mathrm{(D)}\ 188\qquad$ $\mathrm{(E)}\ 210$