Solution:

For the area to be less than the circumference, we must have $$\pi r^2 \lt 2\pi r$$$$r \lt 2.$$

This means the diameter must be less than $4.$ There are three possible rolls that satisfy this: $$(1, 1), (1, 2), (2, 1).$$

The probability is then $$\dfrac{3}{36} = \dfrac{1}{12}.$$

Thus, B is the correct answer.