To calculate the volume of a cone, we 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.
From the problem, we know the following:
- The diameter of the base is 14 millimeters, so the radius \( r \) is half of the diameter:
\[ r = \frac{14}{2} = 7 \text{ millimeters} \]
- The height \( h \) is given as 12 millimeters.
Now, we can substitute these 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 this back into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 49 \times 12 \]
Calculating \( 49 \times 12 \):
\[ 49 \times 12 = 588 \]
Now, substituting this into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 588 \]
Calculating \( 3.14 \times 588 \):
\[ 3.14 \times 588 \approx 1846.32 \]
Now, divide by 3:
\[ V \approx \frac{1846.32}{3} \approx 615.44 \]
Finally, rounding to the nearest hundredth, the volume of the cone is:
\[ \boxed{615.44} \text{ cubic millimeters} \]