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

Responses

9 m3
9 m cubed

3 m3
3 m cubed

36 m3
36 m cubed

4 m3

Answer 1

To understand the volume of a pyramid in relation to the volume of a cube, we can consider their shapes and dimensions.

Let's say we have a cube that has a volume of 12 cubic meters. The formula for the volume of a cube is given by:

\[ \text{Volume of a cube} = \text{side length}^3 \]

Since the volume of the cube is 12 m³, we can find the side length (s) as follows:

\[ s^3 = 12 \] \[ s = \sqrt[3]{12} \]

Now, let’s think about a square-based pyramid that has the same base area and the same height as the cube. The formula for the volume of a square-based pyramid is:

\[ \text{Volume of a pyramid} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

The base area of the pyramid is the same as the area of the cube's base, which is \(s^2\), and the height of the pyramid is equal to the side length of the cube (which is also s).

So, the volume of the pyramid can be expressed as:

\[ \text{Volume of the pyramid} = \frac{1}{3} \times s^2 \times s = \frac{1}{3} \times s^3 \]

Since we already know the volume of the cube is \(s^3 = 12\), we can substitute that into the pyramid's volume formula:

\[ \text{Volume of the pyramid} = \frac{1}{3} \times 12 = 4 \text{ m}^3 \]

Thus, the volume of the square-based pyramid with the same base area and height as the cube is 4 m³.