What is the formula for the volume of a pyramid?

• A. V = 1/3 * base area * height
• B. V = base area * height
• C. V = base area * height / 2
• D. V = (1/3) * height

Answer: (A)

Volume depends on how big the base is and how tall the pyramid is. For a pyramid, the volume is one third of the base area times the height. This comes from comparing the pyramid to a prism with the same base and height—the prism’s volume is base area times height, but a pyramid with that base and height fills only a third of that space. You can also see it by slicing the pyramid with planes parallel to the base; the cross-sectional areas grow with the square of distance from the apex, and integrating up to the full height gives the same 1/3 factor. So the correct combination is base area times height, all divided by three. The other forms either miss the base area, miss the division by three, or rewrite the expression in a way that matches a prism, not a pyramid.