To find the volume of the pyramid, we first need to determine the side length of the cube since we know its volume is 54 cubic units.
The volume \( V \) of a cube is given by the formula:
\[ V = s^3 \]
where \( s \) is the side length of the cube. Setting this equal to 54, we find \( s \):
\[ s^3 = 54 \]
Taking the cube root of both sides gives us:
\[ s = \sqrt[3]{54} \]
Next, since the pyramid has the same base area and height as the cube, we first need to evaluate the base area of the cube. The base area \( A \) of the cube is:
\[ A = s^2 \]
Calculating \( s^2 \):
\[ s^2 = (\sqrt[3]{54})^2 = 54^{2/3} \]
Now, we will compute the height of the cube, which is equal to the side length \( s \) of the cube:
\[ h = s = \sqrt[3]{54} \]
The volume \( V_p \) of a square-based pyramid is given by the formula:
\[ V_p = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Substituting for base area (which is \( s^2 \)) and height \( (s) \):
\[ V_p = \frac{1}{3} \times s^2 \times s = \frac{s^3}{3} \]
Substituting \( s^3 = 54 \):
\[ V_p = \frac{54}{3} = 18 \]
Therefore, the volume of the pyramid is:
\[ \boxed{18} \text{ cubic units} \]