Volume of Solids Calculator

Cube: edge³. Box (cuboid): length × width × height. Cylinder: π × radius² × height — the area of the circular base multiplied by how tall it is. That is the general pattern for anything with a constant cross-section: base area times height. A box is the same idea with a rectangular base. Cone: π × radius² × height ÷ 3. Pyramid: base area × height ÷ 3. Both are the corresponding straight-sided solid divided by three. Sphere: 4/3 × π × radius³. A radius of 3 gives about 113.1 cubic units. Two of these carry a measurement trap. For the cone, height is the vertical height from the base to the tip, not the slant along the outside surface — if you measured the slant, the true height is √(slant² − radius²). For the pyramid, the input is a base area, not a side length, which is what lets the same formula cover square, rectangular, triangular and hexagonal bases alike.

The factor of three is not a coincidence or an approximation. Fill a cone with water and pour it into a cylinder of the same base and height, and it takes exactly three cones to fill it. The same holds for a pyramid inside the matching prism, for any base shape at all. The reason is that a cone tapers linearly, so its cross-section at a given fraction of the height is a scaled copy of the base. Because area scales with the square of the linear factor, the cross-section halfway up has one quarter of the base area, not one half. Adding those shrinking slices up gives one third of the cylinder — the same result as integrating x². A related idea, Cavalieri's principle, says that two solids with the same height and equal cross-sectional area at every level have the same volume. That is why a leaning stack of coins holds exactly as much as a straight one, and why an oblique cylinder has the same volume as a right cylinder of the same height. It is also the reason "height" always means perpendicular height, never the tilted length. The sphere has its own famous ratio. It occupies exactly two thirds of the smallest cylinder that encloses it, a result Archimedes valued highly enough that it was reportedly carved on his tomb.

Volume lives in three dimensions, so it follows the cube of the scale factor. Double every linear dimension and the volume goes up eightfold; triple it and you have 27 times as much. This has consequences that feel wrong until you check them. A pot twice as wide and twice as tall holds eight times as much soup. A storage cube 1.25 times bigger in each direction holds nearly twice the volume. A pipe with twice the diameter carries four times the flow at the same velocity, because flow follows the cross-sectional area rather than the volume. It also means small measurement errors get magnified. A 2% error in each dimension of a box compounds to roughly 6% in the volume — worth knowing before ordering concrete or gravel. The inverse of the rule is just as useful: to double a volume you multiply each dimension by only the cube root of 2, about 1.26. A 26% increase in every direction doubles the capacity. Surface area, meanwhile, only grows with the square. So as a solid gets larger, its surface-to-volume ratio falls — which is why large ice blocks melt more slowly than crushed ice of the same total mass, and why a big pot of stew cools far more slowly than a small one.

Volume becomes a useful number when you convert it into the units a container is actually rated in. In metric this is deliberately painless: 1 m³ = 1,000 litres, and 1 litre = 1,000 cm³, so a cubic centimetre is a millilitre exactly. A tank measuring 0.8 × 0.5 × 1.2 m holds 0.48 m³, which is 480 litres. In US customary units the numbers are less friendly but exact: a US gallon is 231 cubic inches, fixed by definition. Since a cubic foot is 1,728 cubic inches, that makes a cubic foot about 7.48 US gallons. A cubic yard, the unit concrete and mulch are sold in, is 27 cubic feet. Be careful with the word gallon: the US liquid gallon and the imperial gallon used in the UK are different sizes, and so are US and imperial pints and fluid ounces. Recipes and tank specs from different countries are not interchangeable. One more distinction: volume is not capacity. A tank filled to the brim is rarely how a tank is used, and the geometric volume ignores wall thickness, fittings, expansion space and the outlet height. Treat the calculated figure as the theoretical maximum and subtract from there.

Every input for a given solid must be in the same unit, and the result comes out in the cube of that unit. Because volume cubes the length, a unit slip is amplified brutally: mixing centimetres and metres in one calculation is off by a factor of a million, not a hundred. The cubic conversions follow the same rule as the linear ones, cubed. One cubic metre is 1,000,000 cubic centimetres. One cubic foot is 1,728 cubic inches. One cubic metre is about 35.3 cubic feet. When measuring a real container, decide whether you want internal or external dimensions. Capacity needs the inside; shipping and storage space needs the outside. On a thick-walled tank the difference is easily 10%. And for a shape none of these formulas covers, there is a method that always works: displacement. Submerge the object in a container of water and measure how much the level rises. The volume of water displaced is the volume of the object, whatever its shape — the observation Archimedes is remembered for, and still the most reliable way to measure something genuinely irregular.