To find the height of a cylinder when the radius and volume are known, we can use the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius, and
- \( h \) is the height.
From the problem, we know:
- \( V = 441\pi \)
- \( r = 7 \text{ mm} \)
First, we can substitute the known values into the volume formula:
\[ 441\pi = \pi (7^2) h \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Substituting this back into the equation gives:
\[ 441\pi = \pi (49) h \]
To simplify, we can divide both sides of the equation by \( \pi \):
\[ 441 = 49h \]
Now, to solve for \( h \):
\[ h = \frac{441}{49} \]
Calculating \( \frac{441}{49} \):
\[ h = 9 \]
Thus, the height of the cylinder is:
\[ \boxed{9 \text{ mm}} \]