To find the volume of a cone, you can use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius of the base,
- \( h \) is the height of the cone, and
- \( \pi \) is a constant approximately equal to 3.14.
Given the height \( h = 12 \) mm and the diameter of the base \( d = 14 \) mm, we can calculate the radius:
\[ r = \frac{d}{2} = \frac{14}{2} = 7 \text{ mm} \]
Now substitute the values into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times (7^2) \times 12 \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Now substituting that back in:
\[ V = \frac{1}{3} \times 3.14 \times 49 \times 12 \]
Calculating \( 49 \times 12 \):
\[ 49 \times 12 = 588 \]
Now substituting that back in:
\[ V = \frac{1}{3} \times 3.14 \times 588 \]
Calculating \( 3.14 \times 588 \):
\[ 3.14 \times 588 = 1847.92 \]
Now dividing by 3:
\[ V = \frac{1847.92}{3} = 615.9733333 \]
Rounding to the nearest hundredth:
\[ V \approx 615.97 \]
Thus, the volume of the cone is approximately \( \boxed{615.97} \) cubic millimeters.