the volume of a pyramid and cone
A pyramid (a base with triangular sides meeting at a point) and a cone (a circular base tapering to a point) both come to a sharp apex. If you fill an empty cone with sand and pour it into a cylinder of the same base and height, you will need exactly three coneloads to fill the cylinder. That one-third is the heart of the matter.
The volume is V = (1/3) · B · h, where B is the area of the base and h is the perpendicular height from the base up to the apex (measured straight up, not along the slant). For a cone B = pi · r^2, giving V = (1/3) · pi · r^2 · h; for a square pyramid B = s^2, giving V = (1/3) · s^2 · h. In every case the tapering solid holds exactly one third of the prism or cylinder that shares its base and height. The remaining two thirds is the space between the slanted sides and the straight walls.
Why exactly one third, and not some other fraction? It can be proved by slicing a cube into three congruent pyramids, or by Cavalieri's principle comparing cross-sections, both showing the factor is exactly 1/3 for every apex position — the apex may sit off to one side (an oblique cone) and the volume is unchanged, since only the perpendicular height matters. Rigorously pinning down 'exactly' needs a limit argument, which is why the full proof lives in analysis, but the one-third factor itself is solid geometry.
A cone of radius 3 cm and height 4 cm has volume (1/3) · pi · 3^2 · 4 = (1/3)·36·pi = 12·pi ≈ 37.7 cm^3 — exactly one third of the cylinder pi · 3^2 · 4 ≈ 113 cm^3.
V = (1/3) × base × height; a cone is one third of its enclosing cylinder.
The h in the volume is the perpendicular height to the apex, not the slant height; the slant height is longer and is the one used for lateral surface area, so do not swap them.