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)

Responses

9 m3
9 m cubed

3 m3
3 m cubed

4 m3
4 m cubed

36 m3

1 answer

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:

  1. Base Area of the pyramid is \( s^2 \).
  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.