Step 1 :Given the volume of the cylinder is \(162 \mathrm{~cm}^{3}\), and the cone and cylinder have the same height and radius.
Step 2 :The formula for the volume of a cylinder is \(V_{cylinder} = \pi r^2 h\), and the formula for the volume of a cone is \(V_{cone} = \frac{1}{3} \pi r^2 h\).
Step 3 :Since the cone and cylinder have the same height and radius, we can write the volume of the cone as \(V_{cone} = \frac{1}{3} V_{cylinder}\).
Step 4 :Substitute the given volume of the cylinder: \(V_{cone} = \frac{1}{3} (162)\).
Step 5 :Calculate the volume of the cone: \(V_{cone} = 54\).
Step 6 :\(\boxed{54 \mathrm{~cm}^{3}}\) is the volume of the cone.