How is the volume of a pointed figure calculated?

The volume of a pointed figure, which typically refers to a pyramid or a cone, is calculated using the formula V = 1/3Bh. In this formula, B represents the area of the base of the figure, and h represents the height from the base to the apex (the pointed part) of the figure.

The reason for the factor of 1/3 in the formula is that a pointed figure occupies one-third the space of a prism that has the same base area and height. This is evident when visualizing the relationship between cones and cylinders or pyramids and prisms; for every unit of height, the cone or pyramid extends less volume compared to a prism due to its tapering shape. Thus, the formula uniquely accounts for this reduced volume by incorporating the one-third multiplier.

This derivation can be further understood through calculus and the integration of cross-sectional areas, but the essence remains that the pointed shape occupies a reduced volume compared to a solid with a flat base and the same height, justifying the 1/3 coefficient.