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The volume of a pyramid or cone is V = (1/3) * B * h, where B is the base area and h is the perpendicular height. A pyramid or cone is exactly one-third the volume of a prism or cylinder with the same base and height. This 1/3 factor can be demonstrated by filling experiments and proven rigorously using calculus (Cavalieri's principle with varying cross-sections).
Demonstrate the 1/3 relationship physically: fill a cone with water and pour it into a cylinder of the same base and height three times to fill it. Apply the formula to regular pyramids and cones. Practice finding height or radius given the volume.
From your study of prisms and cylinders, you know that their volumes follow the same logic: V = B × h, where B is the base area and h is the perpendicular height. The volume of a prism is just "stack up identical cross-sections from bottom to top." A pyramid and a cone work the same way in spirit — but the cross-sections are not identical. As you move upward in a pyramid, the cross-sections get smaller, tapering toward the apex. This tapering is exactly what introduces the 1/3 factor: V = (1/3) × B × h.
The physical demonstration is the most convincing way to feel this: a cone and a cylinder with the same base and height, where pouring the cone full of water into the cylinder three times exactly fills it. No calculation needed — the 1/3 is built into the geometry. The rigorous proof uses Cavalieri's principle: two solids with the same height have equal volumes if every horizontal cross-section at the same height has the same area. For a pyramid of height h, the cross-section at height z above the base is a scaled-down version of the base with linear scale factor (h-z)/h, so its area is B × ((h-z)/h)². Integrating this from 0 to h gives (1/3)Bh — the calculus confirms what the pouring experiment showed.
Applying the formula requires two steps: identify the base shape and compute its area, then multiply by h and 1/3. For a square pyramid, B = s² where s is the side of the square base. For a cone, B = πr². For a triangular pyramid (tetrahedron), B = (1/2) × base × height of the triangular face. The formula V = (1/3)Bh handles all of these uniformly — the shape of the base goes into B, and the rest is the same.
The most common error is confusing slant height (the distance along the face from base edge to apex) with perpendicular height (the vertical distance from base to apex, measured straight up through the interior). Slant height is relevant for surface area; perpendicular height is what goes into the volume formula. In a right pyramid or cone, the perpendicular height is the leg of a right triangle whose hypotenuse is the slant height and whose other leg is the distance from the center of the base to the edge. You can always recover the perpendicular height using the Pythagorean theorem if only slant height is given.