Question

practice calculating the volume of right and oblique pyramids.
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 $V_{cube} = B \times h$.

Step2: Set total pyramid volume equal to cube volume

Let the height of each pyramid be $h_p$. The volume of one pyramid is $V_{pyramid} = \frac{1}{3} B h_p$. The total volume of 6 pyramids is $6 \times \frac{1}{3} B h_p = 2 B h_p$.

Step3: Solve for $h_p$

Set total pyramid volume equal to cube volume:
$$2 B h_p = B h$$
Cancel $B$ from both sides, then solve for $h_p$:
$$h_p = \frac{h}{2}$$

Answer:

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