To find the volume of a square-based pyramid with the same base area and height as a cube, we can use an informal approach based on the relationship between the volumes of these two shapes.
Let's start with the volume of the cube. The volume \( V \) of a cube is given by the formula:
\[ V = \text{side length}^3 \]
Given that the volume of the cube is 12 cubic meters, we can deduce that the cube has a side length that satisfies this volume.
Now, the pyramid has a square base and the same height as the cube. The volume \( V \) of a square-based pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
The base area of the pyramid is the same as the base area of the cube, and since they share the same height as well, we can position these shapes in relation to one another.
In general, if you take a cube and a pyramid that both have the same base and height, the pyramid will occupy one-third of the volume of the cube. That’s a key relationship between these volumes.
Since the volume of the cube is 12 m³, we can find the volume of the pyramid by applying the volume relationship:
\[ V_{pyramid} = \frac{1}{3} \times V_{cube} \] \[ V_{pyramid} = \frac{1}{3} \times 12 \text{ m}^3 = 4 \text{ m}^3 \]
Thus, the volume of the pyramid is 4 m³.