Mass Calculator

This is a mass calculator based on density and volume. Mass is density multiplied by volume, and the calculator takes and returns a wide range of units, so a density quoted in pounds per cubic foot and a volume in litres still produce an answer.

In the equation, m is mass, ρ is density and V is volume. The SI unit for density is the kilogram per cubic metre, volume is in cubic metres and mass comes out in kilograms. It is a rearrangement of the density equation, so the same relationship also gives density from a known mass and volume.

A cubic metre of copper at 8,900 kg/m³ therefore has a mass of 8,900 kg, which is where the worked example above comes from.

What is mass?

Mass is usually defined as the amount of matter in an object. It is most commonly measured as inertial mass, meaning an object's resistance to acceleration under a net force. Matter itself is loosely defined in science and cannot be measured precisely; in classical physics it is any substance that has mass and volume.

The mass of an object is often correlated with its size, but larger volumes do not always mean more mass. An inflated balloon has far less mass than a golf ball made of silver. The figures are worth seeing: a 30 cm balloon holds about 0.0141 m³ of air, which is 17 grams, while a golf ball of silver occupies 4.07 × 10⁻⁵ m³ and comes to 427 grams. The silver ball is about 350 times smaller and roughly 25 times heavier.

Many units are used for mass around the world. The standard unit under the International System of Units is the kilogram.

Other definitions of mass exist. Active gravitational mass measures how much gravitational force an object exerts; passive gravitational mass measures the gravitational force exerted on an object within a known field. These are conceptually distinct from inertial mass, and no experiment has yet demonstrated a significant difference between gravitational and inertial mass. That equivalence is not a minor detail: it is the observation general relativity is built on.

Mass against weight

The words are used interchangeably in ordinary speech, and even though mass is usually determined by weighing something on a spring scale, they are not equivalent.

Mass stays the same wherever the object is, which makes it an intrinsic property. Weight changes with gravity, because it measures the object's resistance to its natural state of free fall. Gravity on the Moon is roughly one sixth of Earth's, so a person with a mass of 70 kg would weigh about a sixth of their Earth weight there while still having a mass of 70 kg. In numbers, 686 newtons on Earth against 114 on the Moon.

Here F is force, G is the gravitational constant, m₁ and m₂ are the two masses, and r is the distance between their centres. Where the gravitational field is constant, weight is proportional to mass, and using the same units for both causes no trouble.

In the metric system weight is a force measured in newtons:

where g is the acceleration due to gravity, about 9.8 m/s² at the Earth's surface. An object in free fall is weightless however strong the field around it, which is why astronauts in orbit float despite being well inside Earth's gravity. Where an object is accelerated by other forces, as in a centrifuge, weight follows from the total acceleration away from free fall, known as proper acceleration.

Mass itself is defined through:

and the density route above is simply the practical way to get at it when you know what something is made of and how big it is.

The kilogram is no longer a lump of metal

For 130 years the kilogram was defined by a single platinum-iridium cylinder held near Paris, the International Prototype of the Kilogram. Every kilogram in the world was traceable to that object, which was a problem, because the object drifted: comparisons with its official copies showed divergence of tens of micrograms over a century.

That ended in 2019. The kilogram is now defined by fixing the Planck constant at exactly 6.62607015 × 10⁻³⁴ joule seconds, and realising mass through a Kibble balance, which weighs a mass against an electromagnetic force that can be measured in terms of that constant. The practical effect on everyday measurement was nil, deliberately, but the kilogram is now reproducible from physics anywhere rather than borrowed from an artefact in a vault.

One consequence turns up in this calculator. The atomic mass unit is tied to experimentally measured constants rather than defined exactly, so its accepted value shifts slightly as measurements improve. This page uses the 2018 CODATA figure of 1.66053906660 × 10⁻²⁷ kg. Tables still carrying the older 1.660540 × 10⁻²⁷ will disagree in the seventh significant digit, which is why two calculators can differ on an atomic-mass-unit conversion and both look right.

Densities worth knowing

The calculator is only as good as the density you feed it, so these are the common reference figures, all at room temperature and ordinary pressure unless noted.

Water at 1,000 kg/m³ is the anchor worth memorising, because it makes the metric system do the work for you: one litre of water is one kilogram, and one cubic metre is one tonne. In imperial units the same cubic foot of water is 28.32 kg, or 62.43 pounds.

Note that copper is 8,960 rather than the 8,900 in the worked example. Reference densities vary a little between sources depending on alloy, purity and temperature, and 8,900 is a common rounded figure for copper. For engineering work, take the density from the material's own datasheet rather than a general table.

Density is not a fixed property

Two things move it. Temperature expands almost everything, so density falls as things warm: water is densest at 4°C, which is why ice floats and why lakes freeze from the top down. Pressure compresses gases enormously and liquids and solids barely at all, which is why a gas density is meaningless without stating the conditions.

This is why air appears in the table with a temperature and an altitude attached while lead does not. Air at sea level is 1.225 kg/m³; at 10,000 metres it is around a third of that, and the same volume of it has a third of the mass.

Common mistakes

The first is mixing unit systems inside one calculation. A density in pounds per cubic foot with a volume in cubic metres gives a meaningless number unless one of them is converted. The calculator handles this by attaching a unit to each field, so check both dropdowns rather than just the numbers.

The second is confusing mass with weight in a context where it matters. A kitchen scale reads mass in grams by assuming Earth's gravity; the same scale on the Moon would under-read by a factor of six while the food would be unchanged.

The third is treating a table density as exact. Alloy, purity, temperature and porosity all shift it, and for concrete or timber the spread between samples is wide enough to dominate any rounding in the arithmetic.

The fourth is forgetting that volume scales as the cube of a length. Doubling every dimension of an object multiplies its volume, and so its mass, by eight.