Describe an informal argument for the volume of a pyramid based on the volume of a cube. A cube has a volume of 12 cubic meters. A square-based pyramid has the same base area and the same height as the cube. What is the volume of the pyramid? (1 point)

Question

• 36 m3
• 9 m3
0 4m3
• 3 m3

Answer 1

To informally argue for the volume of a pyramid based on the volume of a cube, we can start by examining the relationships between the two shapes.

1. Understanding the Cube:
   A cube has a volume calculated by the formula \( V = \text{side}^3 \). If we know that the volume of our cube is 12 cubic meters, then we can find the dimensions of the cube. Let's say the side length of the cube is \( s \). Therefore, we have: \[ s^3 = 12 \text{ m}^3 \] The cube has equal sides.

2. Base of the Pyramid:
   A square-based pyramid shares the same base area as the top face of the cube. The base area of the pyramid is \( s^2 \), which equals the area of the square face of the cube (because it has the same side lengths). We can calculate the base area from the cube’s volume: \[ \text{Base Area} = s^2 \]

3. Height of the Pyramid:
   Since the pyramid has the same height as the cube, the height of the pyramid is \( s \), which is also the side length of the cube.

4. Volume of the Pyramid:
   The volume of a pyramid is given by the formula: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Substituting the base area and height of our pyramid, we have: \[ V = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3 \] We already know from the volume of the cube that \( s^3 = 12 \text{ m}^3 \), so we can replace \( s^3 \): \[ V = \frac{1}{3} \times 12 \text{ m}^3 = 4 \text{ m}^3 \]

5. Conclusion:
   Therefore, the volume of the pyramid, which has the same base area and height as the cube, is \( 4 \) cubic meters.

From the options given, the correct answer is:

• 4 m³.