To find the volume of the pyramid, we first need to understand the relationship between the pyramid and the cube.
The volume of a cube (V_cube) is given by the formula:
\[ V_{cube} = s^3 \]
where \(s\) is the side length of the cube. Given that the volume of the cube is 54 cubic units, we have:
\[ s^3 = 54 \] \[ s = \sqrt[3]{54} \approx 3.78 \text{ units} \]
Since the cube has height (h) equal to its side length \(s\), the height of the cube is also approximately 3.78 units.
Next, we know that the square-based pyramid has the same height and base area as the cube. The height of the pyramid is also \(h\) (the same as the cube), and the base area (A_base) of the pyramid is equal to the base area of the cube.
The base area of the cube can be calculated as:
\[ A_{base} = s^2 = \left(\sqrt[3]{54}\right)^2 \] \[ A_{base} = 54^{2/3} \]
However, for the pyramid, we need to calculate its volume. The volume of a square-based pyramid (V_pyramid) is given by the formula:
\[ V_{pyramid} = \frac{1}{3} \times A_{base} \times h \]
Using the height h (which is \(s\), since they both are equal):
\[ V_{pyramid} = \frac{1}{3} \times A_{base} \times s \]
Since \(A_{base} = s^2\):
\[ V_{pyramid} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3 \]
From our earlier calculations, we know \(s^3 = 54\). Therefore:
\[ V_{pyramid} = \frac{1}{3} \times 54 = 18 \text{ cubic units} \]
Thus, the volume of the pyramid is:
\[ \boxed{18} \text{ cubic units} \]
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