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.
Given:
- The diameter of the base is 14 millimeters, so the radius \( r \) is half of that:
\[ r = \frac{14}{2} = 7 \text{ mm} \]
- The height \( h \) is given as 12 millimeters.
Now, substituting 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 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 value back into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 588 \]
Calculating \( 3.14 \times 588 \):
\[ 3.14 \times 588 = 1847.92 \]
Now divide by 3:
\[ V = \frac{1847.92}{3} \approx 615.9733 \]
Rounding to the nearest hundredth:
\[ V \approx 615.97 \text{ cubic millimeters} \]
Therefore, the volume of the cone is approximately:
\[ \boxed{615.97} \text{ cubic millimeters} \]