To find the volume of a right cone, we use the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base, and \( h \) is the height of the cone.
We are given:
- The diameter of the base is 14 mm, so the radius \( r \) is:
\[ r = \frac{14}{2} = 7 \text{ mm} \]
- The height \( h \) is 12 mm.
Now, we can substitute \( r \) and \( h \) into the volume formula. First, we compute \( r^2 \):
\[ r^2 = 7^2 = 49 \]
Now, substituting the values 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 back into the volume calculation:
\[ V = \frac{1}{3} \times 3.14 \times 588 \]
Calculating \( 3.14 \times 588 \):
\[ 3.14 \times 588 = 1846.32 \]
Now, dividing by 3:
\[ V = \frac{1846.32}{3} \approx 615.44 \]
Therefore, the volume of the cone is approximately:
\[ \boxed{615.44} \text{ cubic millimeters} \]