To find the volume of the square-based pyramid, we start with the information given:
- The volume of the cube is 9 cubic meters.
- The height of the pyramid is equal to the side length of the cube.
- The base area of the pyramid is equal to the area of the base of the cube.
First, since the volume of a cube is calculated as: \[ V = s^3 \] where \( s \) is the side length, we have: \[ s^3 = 9 \quad \Rightarrow \quad s = \sqrt[3]{9} \approx 2.08 , \text{m} \]
Next, we can calculate the base area of the cube. The base area (which is a square) is: \[ \text{Base Area} = s^2 = (\sqrt[3]{9})^2 = 9^{2/3} , \text{m}^2 \]
Now, we can find the volume of the pyramid. The volume \( V \) of a square-based pyramid is given by: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Given that:
- Base Area = \( 9^{2/3} \)
- Height = \( s = \sqrt[3]{9} \)
We can substitute these values into the formula for the volume of the pyramid: \[ V = \frac{1}{3} \times 9^{2/3} \times \sqrt[3]{9} \]
Now, we simplify: \[ = \frac{1}{3} \times 9^{2/3} \times 9^{1/3} = \frac{1}{3} \times 9^{(2/3 + 1/3)} = \frac{1}{3} \times 9^{1} = \frac{9}{3} = 3 \]
Thus, the volume of the pyramid is: \[ \boxed{3 , \text{m}^3} \]