Q1: What happens to the volume if the height of a pyramid doubles?
A1: Because perpendicular height ($h$) has a linear relationship within the formula $V = \frac{1}{3}Bh$, doubling the height while keeping the base unchanged doubles the total volume. In contrast, doubling the base edge length quadruples the volume, because the base area scales quadratically ($2^2 = 4$).
Q2: Can the same formula work for irregular or oblique pyramids?
A2: Yes. Under Cavalieri's Principle, if two three-dimensional solids share identical base areas and equal heights across every parallel cross-section, they possess identical volumes. Even if the apex leans heavily to one side (an oblique pyramid), the formula $V = \frac{1}{3}Bh$ holds true as long as you measure height strictly along a perpendicular line from the apex to the plane of the base.
Q3: How do you find the volume of a truncated pyramid (frustum)?
A3: A frustum forms when the top of a pyramid is sliced off parallel to its bottom. Calculate the volume using the formula $V = \frac{1}{3}h(B_1 + B_2 + \sqrt{B_1 B_2})$, where $h$ is the vertical height separating the cuts, $B_1$ is the lower base area, and $B_2$ is the upper base area.
Q4: Why does a cone use the exact same one-third multiplier?
A4: A cone is geometrically a pyramid with a circular base. As a regular polygon gains infinite sides, it approximates a circle. Thus, the equation $V = \frac{1}{3}Bh$ becomes $V = \frac{1}{3}\pi r^2 h$.
