Question

six identical square pyramids can fill the same volume as a cube with the same base. if the height of the cube is h units, what is true about the height of each pyramid?
the height of each pyramid is \\(\frac{1}{2}h\\) units.
the height of each pyramid is \\(\frac{1}{3}h\\) units.
the height of each pyramid is \\(\frac{1}{6}h\\) units.
the height of each pyramid is h units.

Explanation:

Step1: Define cube volume

Let the base area of the cube (and each pyramid) be $B$. The volume of the cube is $V_{\text{cube}} = B \times h$.

Step2: Set total pyramid volume equal

Total volume of 6 pyramids equals cube volume: $6 \times V_{\text{pyramid}} = B h$.

Step3: Substitute pyramid volume formula

The volume of one square pyramid is $V_{\text{pyramid}} = \frac{1}{3} B h_p$, where $h_p$ is the pyramid's height. Substitute:
$$6 \times \frac{1}{3} B h_p = B h$$

Step4: Simplify to solve for $h_p$

Simplify left side: $2 B h_p = B h$. Divide both sides by $B$: $2 h_p = h$, so $h_p = \frac{1}{2} h$.

Answer:

The height of each pyramid is $\frac{1}{2} h$ units.