The circumference of a circle can be calculated using the formula:
\[ C = 2 \pi r \]
where \( C \) is the circumference and \( r \) is the radius of the circle.
For circle A, with a radius of 6 millimeters:
\[ C_A = 2 \pi \times 6 \approx 12 \pi \approx 37.68 \text{ mm} \]
which is approximately 38 millimeters, as given.
For circle B, with a radius of 60 millimeters:
\[ C_B = 2 \pi \times 60 = 120 \pi \]
Approximating \( \pi \) as 3.14, we get:
\[ C_B \approx 120 \times 3.14 = 376.8 \text{ mm} \]
Thus, the approximate circumference of circle B is:
\[ \boxed{376.8} \text{ mm} \]