What is the formula for the volume of a pyramid or cone?

The formula for the volume of a pyramid or cone is expressed as ( V = \frac{1}{3}Bh ), where ( B ) represents the area of the base and ( h ) represents the height of the pyramid or cone. This formula is derived from the principle that the volume of a pyramid or cone is one-third of the product of the area of the base and the height.

This means that if you calculate the area of the base and then multiply it by the height, the resulting value represents the volume of a prism with the same base and height. Since a pyramid or cone tapers to a point, the actual volume is only a third of that, hence the ( \frac{1}{3} ) factor in the formula.

Understanding the dimensional relationship in three-dimensional space helps to grasp why the volume diminishes to one-third: it accounts for the shape's gradual convergence from the base to its apex, compared to a solid prism that maintains a constant cross-section. This makes option A the correct representation of the volume for both pyramids and cones.