Volume Formulas
Every volume formula multiplies a base area by a height, or a close variant of that idea. A box and a cylinder both compute base area times height directly; a cone and a pyramid use exactly one-third of that, because a cone or pyramid tapers to a point rather than holding its cross-section constant all the way up.
Formulas
Box: V = l × w × h
Cylinder: V = πr²h
Sphere: V = (4/3)πr³
Cone: V = (1/3)πr²h
Pyramid: V = (1/3) × base area × h
The one-third factor for a cone or pyramid is not a rough approximation — it follows exactly from integrating the changing cross-sectional area from base to apex, and it holds regardless of the base shape, which is why the pyramid formula uses base area rather than a shape-specific expression.
Why Volume Grows So Fast With Size
Length scales linearly, area scales with the square, and volume scales with the cube — so doubling every dimension of a solid multiplies its volume by eight, not two or four. This single fact explains why a slightly larger container holds dramatically more than it looks like it should.
Doubling a cube's side multiplies volume by 8
A 2-inch cube holds 8 cubic inches; a 4-inch cube holds 64. The side only doubled, but the volume rose eightfold, because 2³ vs 4³ compares as 1:8.
This is why packaging upsizes look modest but hold much more
A container that looks 25% bigger in each dimension holds roughly 95% more volume (1.25³ ≈ 1.95), which is a common reason a "large" size at a store offers a much better price per unit than the size difference alone suggests.
Surface area does not keep pace with volume
Surface area scales with the square while volume scales with the cube, so larger objects have proportionally less surface area per unit of volume. This is why large animals struggle to shed heat efficiently and why large batches of material take longer to cool than small ones.
Converting Between Volume Units
Volume units carry the same cubing relationship as the length units they are built from — a factor that trips people up more than any other unit conversion, since the instinct is to convert linearly.
A cubic metre holds exactly 1,000 litres, since a litre is one cubic decimetre and 10 decimetres fit in a metre in each of three dimensions (10³ = 1,000). A US gallon is 3.785 litres, and a cubic foot is 28.317 litres — useful reference points for estimating tank, container, or aquarium capacity without a full unit conversion each time.