Which statement is true about pyramid volume relative to base area B and height h?

Which statement is true about pyramid volume relative to base area B and height h?

• A. The volume of a pyramid is twice base area times height.
• B. The volume of a pyramid is one third of base area times height.
• C. The prism volume is one third of base area times height.
• D. The pyramid volume equals base area times height.

Answer: (B)

The key idea is that a pyramid’s volume is determined by the base area and the height, but it’s one third as large as the rectangular prism that has the same base and height. So for a base area B and height h, the volume is V = (1/3) B h. Think of slicing the pyramid with horizontal cross-sections. At the very bottom, the cross-section matches the base area B. As you go up toward the apex, every linear dimension scales down linearly to zero, so the cross-sectional area scales like the square of that linear scale, ending up proportional to (1 − t/h)^2 at height t. If you add up all those slices from bottom to top, you’re integrating B(1 − t/h)^2 from t = 0 to t = h, which gives Bh/3. Equivalently, the prism with the same base and height has volume Bh, and a pyramid sits inside that prism in such a way that its volume is exactly one third of Bh. This is why the correct statement is that the pyramid’s volume equals one third of base area times height. The other options would correspond to a twofold multiple or to the prism volume, not the pyramid.

The key idea is that a pyramid’s volume is determined by the base area and the height, but it’s one third as large as the rectangular prism that has the same base and height. So for a base area B and height h, the volume is V = (1/3) B h.

Think of slicing the pyramid with horizontal cross-sections. At the very bottom, the cross-section matches the base area B. As you go up toward the apex, every linear dimension scales down linearly to zero, so the cross-sectional area scales like the square of that linear scale, ending up proportional to (1 − t/h)^2 at height t. If you add up all those slices from bottom to top, you’re integrating B(1 − t/h)^2 from t = 0 to t = h, which gives Bh/3.

Equivalently, the prism with the same base and height has volume Bh, and a pyramid sits inside that prism in such a way that its volume is exactly one third of Bh. This is why the correct statement is that the pyramid’s volume equals one third of base area times height. The other options would correspond to a twofold multiple or to the prism volume, not the pyramid.