smc

Through inspection, we see that the two digit number $33^2=1089=4$ digits. Notice that any number that has the form $33abcdefg.......$ multiplied by another $33qwertyu.........$ will have its number of digits equal to the sum of the original numbers' digits. In this case, we see that the first number has $19$ digits, and the second number has $15$ digits. Note: this applies for numbers $33--->99$ Hence, the answer is $19+15=34$ digits $⟹\boxed{34}$