Question

which lengths can be used, directly or indirectly, to calculate the volume of the hexagonal right pyramid? select three options. xy and st vu and tw xs and xw tx and wx vu and yz

Explanation:

Step1: Recall pyramid volume formula

The volume of a right pyramid is $V = \frac{1}{3} \times B \times h$, where $B$ is the area of the hexagonal base, and $h$ is the height of the pyramid (distance from apex $T$ to base, i.e., $ST$).

Step2: Analyze each option

Option 1: XY and ST

• $XY$ is the side length of the regular hexagon base. The area of a regular hexagon is $B = \frac{3\sqrt{3}}{2}s^2$, where $s=XY$. $ST$ is the height $h$. We can calculate $V$ directly.

Option 2: VU and TW

• $VU$ is the distance between two opposite vertices of the hexagon (diameter of the circumscribed circle, $d=VU$). For a regular hexagon, side length $s = \frac{d}{2}$, so we can find $B$. $TW$ is a lateral edge; using Pythagoras ($TW^2 = h^2 + (\text{distance from } S \text{ to } W)^2$), since $SW = \frac{VU}{2}$, we can solve for $h=ST$. We can calculate $V$.

Option 3: XS and XW

• $XS$ is half the distance from $X$ to the center along the base (apothem-related, but not enough to find side length alone). $XW$ is a side of the hexagon? No, $XW$ is an edge of the base but we can't get the height of the pyramid from these two lengths. We cannot calculate $V$.

Option 4: TX and WX

• $WX$ is the side length of the hexagon, so we can find $B$. $TX$ is a lateral edge; using Pythagoras ($TX^2 = h^2 + (\text{distance from } S \text{ to } X)^2$), since the distance from center $S$ to vertex $X$ equals the side length $WX$, we can solve for $h=ST$. We can calculate $V$.

Option 5: VU and YZ

• $VU$ is the diameter of the circumscribed circle, so we can find the side length $s=\frac{VU}{2}$ and calculate $B$. But we have no information about the height $h$ of the pyramid. We cannot calculate $V$.

Answer:

• XY and ST
• VU and TW
• TX and WX