Volume units

Litres, millilitres, cubic centimetres, cubic metres — all of it is one quantity, written in different ways. The difficulty sits in a single place: volume conversions are not what intuition trained on lengths suggests.

The cube of edge 10 cm

The whole ladder of units rests on one calculation:

An edge of $10\,\text{cm}$ is $1\,\text{dm}$, so the same cube has a volume of $1\,\text{dm}^3$. And since the litre was defined as the volume of precisely that cube:

From which follows a second, even handier identity — divide both sides by a thousand:

This is neither an approximation nor a school convention: a cubic centimetre and a millilitre are the same volume. A syringe graduated in millilitres measures cubic centimetres.

Why a thousand and not ten

For lengths the factor from $\text{cm}$ to $\text{dm}$ is ten. For volumes it is a thousand. The reason is the one that squares the factor for areas:

Volume comes from multiplying three lengths, so each of them grows $k$ times and the factor enters three times over.

The same rule in three versions — worth carrying in your head as one rule rather than three:

The ladder of units

Every rung is a multiplication or a division by 1000:

In practice two steps are enough to remember: going up multiply by 1000, going down divide by 1000. Multiplying by a thousand moves the decimal point three places to the right, dividing moves it three to the left.

Which unit is used where

• millilitres — medicine, seasoning, cosmetics, small lab glassware;
• litres — drinks, a fuel tank, a watering can, the capacity of an aquarium;
• cubic metres — mains water on the bill, gas, concrete, the volume of a room;
• cubic centimetres — engine capacity, and volumes worked out from geometric formulas.

Converting inside a problem

Dimensions given in different units have to be brought to one unit before the volume is computed, not after. Converting the result then needs a cubic factor (a thousand or a million), while converting the data needs the ordinary length factor.

This lesson is about where those factors come from. When you simply need a value — especially in units outside SI, such as gallons, quarts or fluid ounces — use our volume converter.

Exercises

Type the answer together with the target unit, the one after the arrow — for example 12000 ml. Write the number without spaces, and use a dot for the decimal point.

Common mistakes

• Multiplying by 100 instead of 1000 — the length factor is raised to the third power, not the second.
• $1\,\text{m}^3 = 100\,\text{cm}^3$ — it is a million; this is the costliest slip in the topic.
• Confusing $\text{cm}^3$ with $\text{cm}^2$ — the cube belongs to volume, the square to area.
• Treating a millilitre as a gram — they measure different quantities; the identity holds for water only, and only approximately.
• Converting the result instead of the data — if the dimensions are in mixed units, bring them to one before multiplying.
• Moving the decimal point one place — multiplying by 1000 moves it three.