The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
For a cylinder sharing the same radius and height as the cone, the volume \( V_c \) is given by the formula:
\[ V_c = \pi r^2 h \]
To find the volume of the cylinder, we can relate it to the volume of the cone. Since the volume of the cone is given as 27 mm\(^3\):
\[ V = 27 \text{ mm}^3 \]
From the volume formula for the cone, we can rewrite it in terms of \( \pi r^2 h \):
\[ \frac{1}{3} \pi r^2 h = 27 \]
Multiplying both sides by 3 gives:
\[ \pi r^2 h = 3 \times 27 \]
Calculating that:
\[ \pi r^2 h = 81 \text{ mm}^3 \]
Thus, the volume of the cylinder, which is equal to \( \pi r^2 h \), is:
\[ V_c = 81 \text{ mm}^3 \]
Therefore, the volume of the cylinder that shares the same radius and height as the cone is:
\[ \boxed{81 \text{ mm}^3} \]