Ahmad constructs a square-based pyramid and a cube out of newspaper.

The two figures have the same height and the same base area. What is the volume of the pyramid if the volume of the cube is 54 cubic units? (1 point)
• 90 cubic units
• 18 cubic units
• 6 cubic units
• 36 cubic units

1 answer

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:

\[ \boxed{18} \text{ cubic units} \]