Volume of Solids

If a solid has the same cross-section all the way up, its volume is simply the base area times the height:

V = Bh

That covers every prism and every cylinder. For a cylinder, B = πr², so V = πr²h.

Anything that narrows to a point holds exactly one third as much:

V = ⅓Bh

for pyramids and cones. Same base, same height, one third the volume — and that is worth remembering because it is easy to check against intuition.

The sphere is its own case: V = (4/3)πr³.

Use the *vertical* height here, not the slant height. Slant height belongs to surface area; volume always uses the perpendicular height.

A slanted (oblique) solid has the same volume as an upright one with the same base and height.

For any solid with congruent parallel cross-sections, V = Bh. Cavalieri's principle generalises this: two solids of equal height whose cross-sections have equal area at every level have equal volume, which is why an oblique prism matches the corresponding right prism.

Pyramids and cones satisfy V = ⅓Bh. The factor arises because the cross-sectional area shrinks quadratically with height, and ∫₀ʰ B(1 − x/h)² dx = Bh/3. Three congruent pyramids assemble into a cube, which is the standard non-calculus demonstration.

The sphere's V = (4/3)πr³ follows from Cavalieri applied to a hemisphere and a cylinder with an inscribed cone removed.

Volume uses the perpendicular height throughout; slant height appears only in lateral surface area.

Under a similarity of ratio k, volume scales by k³ while surface area scales by k². The resulting fall in surface-area-to-volume ratio as objects grow is why large animals overheat less easily and why crushed ice melts faster than a block.