Mass, volume, and density are one triangle. Convert the units first, calculate second, and each unknown follows from the other two — with water as the reference every density is read against.

The triangle: mass, volume, and density

Mass, volume, and density are one relationship with three faces, in the same way speed, distance, and time are one relationship with three faces. Mass equals volume times density. Density equals mass divided by volume. Volume equals mass divided by density. Cover the quantity you want and the other two show the operation, and the only trap is the units: the three quantities must be in matching units before the arithmetic runs. Density in kilograms per cubic metre needs volume in cubic metres and gives mass in kilograms. Density in grams per cubic centimetre needs volume in cubic centimetres and gives mass in grams. Density in pounds per cubic foot needs volume in cubic feet and gives mass in pounds. Convert first, calculate second, and the triangle never lies. Mix the units, and it answers precisely and wrongly, which is the worse outcome.

Density itself is a property of the material, not of the object. A litre of water and a swimming pool of water have the same density, because density is mass per unit volume, and doubling the volume doubles the mass with it. That is why density identifies materials: it is the one quantity in the triangle that does not change when the object gets bigger or smaller, as long as the material and the conditions stay the same. It is also why the triangle is useful in both directions: measure the mass and the volume of an unknown object, divide, and the density narrows the material down; know the material and one of the other two quantities, and the third follows without measuring it.

Reference densities: water, concrete, gold, and the rest

Water is the reference, because its density makes the arithmetic transparent. Water has a density of 1 kilogram per litre, which is 1,000 kilograms per cubic metre, which is 1 gram per cubic centimetre, which is 62.4 pounds per cubic foot. A litre of water weighs a kilogram. A cubic foot of water weighs 62.4 pounds. Those are the same fact in two unit systems, and they are the figures the Volume Calculator guide already uses: the 72-litre fish tank whose water weighs 72 kilograms, and the 35.34-cubic-foot capsule displacing water that supports about 2,205 pounds, which is 35.34 times 62.4. Every other material in this guide is read against water, as a multiple of it: denser than water sinks in it, less dense floats, and the multiple says by how much.

Concrete, at about 2,400 kilograms per cubic metre, is 2.4 times as dense as water, which is why a cubic metre of concrete weighs about 2,400 kilograms and a litre of it weighs about 2.4. Gold, at 19,300 kilograms per cubic metre, is 19.3 times as dense as water, which is why the classic demonstration — a litre of gold weighing 19.3 kilograms — feels so disproportionate in the hand. Steel is about 7,850 kilograms per cubic metre, aluminium about 2,700, ice about 917, and air about 1.2 at ordinary conditions. The table below collects the typical figures, each labelled typical because real materials vary: concrete varies with its mix and aggregate, steel with its alloy, wood enormously with species and moisture, and every liquid with temperature. A density quoted without “typical” or “about” is a density overstated, and a density used without checking the conditions is a density misapplied.

Finding mass: volume times density, worked

Water first, because the arithmetic is transparent. A tank holding 72 litres of water holds 72 kilograms of water, at 1 kilogram per litre, before glass and stand — the guide’s own example, and the one to check every other calculation against. A tank holding 250 litres holds 250 kilograms. A cubic metre of water — 1,000 litres — weighs 1,000 kilograms, which is a tonne, and the tonne is defined by exactly that relationship. Concrete next, because the numbers are large enough to matter. A slab 2 cubic metres in volume, at 2,400 kilograms per cubic metre, weighs 4,800 kilograms. A footing of 0.5 cubic metres weighs 1,200 kilograms. The weight follows the volume directly, which is why the concrete weight questions on the Concrete Calculator page are volume questions with density applied, and why a small error in the measured volume becomes a large error in the weight. Gold last, because the numbers surprise. A litre of gold weighs 19.3 kilograms, and half a litre weighs 9.65 kilograms — a small object with the weight of a large one, which is the whole point of density as a concept. The method is identical in all three: volume in matching units, times density, equals mass, with the unit of the answer following from the units of the inputs.

The imperial version is the same calculation in different units. A cubic foot of water weighs 62.4 pounds, so 10 cubic feet weigh 624 pounds. A cubic foot of concrete, at 149.8 pounds per cubic foot, weighs about 150 pounds, which is the figure the concrete weight guide rounds to. A cubic yard of concrete — 27 cubic feet — weighs about 4,050 pounds, which is the hub guide’s figure, and the chain from density to yard weight is the same chain: volume times density, with the units converted at each step. The most common imperial error is the missing conversion: pounds per cubic foot times cubic yards, without dividing by 27, gives a figure 27 times too large, and the error is invisible until the trailer is loaded.

Finding volume: mass divided by density, worked

Volume from mass is the rearrangement people use least, and it is the one that answers the most practical questions: how much space will this mass occupy, how large a container will it need, how much of it can a vehicle carry. A gold bar weighing 19.3 kilograms, at 19,300 kilograms per cubic metre, occupies 19.3 divided by 19,300, which is 0.001 cubic metres, or 1 litre — the guide’s example, and the check that the rearrangement works. A tonne of concrete, 1,000 kilograms at 2,400 kilograms per cubic metre, occupies 0.417 cubic metres. A tonne of water occupies exactly 1 cubic metre. The same tonne, three materials, three volumes, and the density is the whole difference.

The container version of the same problem is worth stating, because it is where the rearrangement is used daily. A container’s capacity is a volume; a delivery’s weight limit is a mass; and density converts one to the other. A tank rated for 1,000 litres holds 1,000 kilograms of water, 2,400 kilograms of concrete if it could hold concrete, and 19,300 kilograms of gold if it could hold gold — and the fact that no tank is rated for all three is the point, because the weight limit, not the volume, is usually the binding constraint for dense materials. Divide the weight limit by the density and the volume that can actually be carried follows, and a load planned by volume alone for a dense material is a load that overloads.

Finding density: measure, divide, and identify with care

Density from mass and volume is a measurement, not a lookup: weigh the object, measure its volume, divide. A block measured at 5 kilograms with a volume of 0.002 cubic metres has a density of 5 divided by 0.002, which is 2,500 kilograms per cubic metre — denser than water, a little denser than concrete, and consistent with stone or dense ceramic rather than with metal or with wood. The identification is a narrowing, not a proof: many materials share a density, mixtures and porous materials have average densities that match no pure material, and the measurement’s own errors — the volume of an irregular object is hard to measure — move the result. Archimedes’ method, measuring volume by water displacement, is the classic remedy for irregular shapes: submerge the object, measure the water displaced, and the displaced volume is the object’s volume, from which density follows by division. The story of the crown and the bath is the story of this method, and it remains the most reliable way to measure the volume of an awkward shape without a formula for it.

Average density deserves its own paragraph, because it is where density reasoning most often misleads. A ship floats although steel sinks, because the ship’s average density — steel plus air-filled spaces, divided by total volume — is less than water’s. A hollow ornament and a solid one of the same material have different average densities. A mixture’s density is the weighted average of its components, and a porous material’s density depends on whether its pores are counted as part of the volume. Density identifies a material only when the object is the material, through and through. When it is an assembly, a mixture, or a hollow form, density describes the assembly, and the identification must be made another way.

Floating, sinking, and why ships and ice both float

An object floats if its average density is less than the fluid’s density, and sinks if it is greater — the buoyancy rule the Volume Calculator guide states for its capsule example. Ice, at about 917 kilograms per cubic metre, floats in water at 1,000, with about 92 percent of its volume submerged and 8 percent above the surface, which is why icebergs show so little and hide so much. A steel ship floats because its average density, counting the air inside it, is less than water’s, and it sinks when that average rises — when compartments flood, for example — past the threshold. A hot-air balloon rises because the heated air inside it is less dense than the surrounding air, and a helium balloon rises for the same reason with a different gas. The rule is one line, and the examples are the whole of naval architecture, glaciology, and ballooning, because each is the rule applied to a different average density.

Temperature, pressure, and the density that moved

Density changes with temperature and pressure, because volume changes with them while mass does not. Liquids expand as they warm, so their density falls: water at 20 degrees is a little less dense than water at 4 degrees, where its density peaks, and the difference matters in precise work, in hydrometry, and in any calculation that quotes a density without a temperature. Gases change far more: air’s density falls sharply with temperature and rises with pressure, which is why a density quoted for air without temperature and pressure is approximate at best, and why the table above says “ordinary conditions” rather than pretending to a precision the figure does not have. Solids change least, but they change: a steel rail expands in summer, and its density falls by the same fraction its volume rises. The discipline is to state the conditions with the density, to use a density measured at the working temperature where precision matters, and to treat every density in a table as a typical value at stated or ordinary conditions, not as a constant of nature.

Mistakes: weight, volume, and the unit that was not converted

First, confusing weight and volume: a litre of feathers and a litre of lead have the same volume and wildly different masses, and the sentence “a litre weighs a kilogram” is true only of water. Second, the unit mismatch: density in kilograms per cubic metre with volume in litres, or pounds per cubic foot with volume in cubic yards, without the conversion that the triangle requires. Third, the average-density error: identifying a hollow, mixed, or porous object by a density that describes the assembly, not the material. Fourth, the temperature error: using a density quoted at one temperature for work at another, where precision matters. Fifth, the precision error: quoting a density to four figures from a measurement good to two, and believing the extra figures. The remedy for all five is the same as for the speed triangle: state the units, convert first, calculate second, and check the answer against water, which is the reference every other density is read against.

Frequently asked questions

How do I find density if I know mass and volume?

Divide mass by volume, in matching units. A block of 5 kilograms with a volume of 0.002 cubic metres has a density of 5 divided by 0.002, which is 2,500 kilograms per cubic metre. State the units with the answer, because the same material reads 2,500 in kilograms per cubic metre and 2.5 in grams per cubic centimetre. For an irregular object, measure the volume by water displacement and divide the mass by the displaced volume.

How do I find volume if I know mass and density?

Divide mass by density, in matching units. A gold bar weighing 19.3 kilograms, at a density of 19,300 kilograms per cubic metre, occupies 19.3 divided by 19,300, which is 0.001 cubic metres, or 1 litre. A tonne of water occupies exactly 1 cubic metre, and a tonne of concrete at 2,400 kilograms per cubic metre occupies 0.417 cubic metres. Convert the units first, and the division does the rest.

How do I find mass if I know volume and density?

Multiply volume by density, in matching units. A tank holding 72 litres of water holds 72 kilograms of water, at 1 kilogram per litre. A slab of 2 cubic metres of concrete, at 2,400 kilograms per cubic metre, weighs 4,800 kilograms. A cubic foot of water weighs 62.4 pounds, so 10 cubic feet weigh 624 pounds. The unit of the answer follows from the units of the inputs.

Why does the same volume weigh different amounts for different materials?

Because density differs. Volume measures space, and density measures how much mass sits in each unit of that space, so the same litre holds 1 kilogram of water, about 2.4 kilograms of concrete, and 19.3 kilograms of gold. Mass equals volume times density, and density is the part that changes. A litre of feathers and a litre of lead have the same volume and wildly different masses for exactly this reason.

Does density change with temperature?

Yes, because volume changes with temperature while mass does not. Liquids expand as they warm, so their density falls, and water is densest at about 4 degrees. Gases change far more: air density falls sharply with temperature and rises with pressure. A density quoted without its temperature and pressure is approximate, and precise work must use a density measured at the working temperature.

Why do ships float if steel sinks?

Because floating depends on average density, not on the density of the material alone. A ship’s average density, counting the steel plus the air-filled spaces divided by the total volume, is less than water’s density of 1,000 kilograms per cubic metre, so it floats. A solid steel block has an average density of about 7,850 and sinks. Flood the compartments, raise the average past the threshold, and the ship sinks: the rule is the same in both cases.