To informally understand the volume of a pyramid based on the volume of a cube, let’s start with what we know about the volume of a cube.
If a cube has a volume of 12 cubic meters, that means it has equal side lengths. The formula for the volume of a cube is given by:
\[ \text{Volume} = \text{side}^3 \]
If we denote the side length of the cube as \( s \), then:
\[ s^3 = 12 \]
Now, if we consider a square-based pyramid that has the same base area and height as the cube, we need to look at how the volumes compare.
The base area of the pyramid (which is a square) is the same as the area of one face of the cube. So, if the side length of the cube is \( s \), the base area of the pyramid is:
\[ \text{Base Area} = s^2 \]
The volume of a pyramid is given by:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Since the pyramid has the same height as the cube, the height \( h \) of the pyramid is equal to the side length of the cube (assuming the cube stands on one of its faces). Thus, \( h = s \).
Now we can substitute the base area and height into the volume formula for the pyramid:
- Base Area of the pyramid is \( s^2 \).
- Height of the pyramid is \( s \).
So, the volume of the pyramid can be calculated as follows:
\[ \text{Volume of Pyramid} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3 \]
Since we know that \( s^3 = 12 \) (the volume of the cube), we can substitute this into our equation for the pyramid's volume:
\[ \text{Volume of Pyramid} = \frac{1}{3} \times 12 = 4 \text{ m}^3 \]
Thus, the volume of the pyramid is 4 m³.