Volume describes how much space a figure occupies in three-dimensional space. It is used, for example, to indicate how much water a container can hold or how much air there is in a room. The unit of volume in the SI system is the cubic metre \( \text{m}^3 \).

 

 

Common volume units

Today, the cubic metre is used as the standard unit, but there are many practical subunits and multiples that are easier to use depending on the situation.

 

Unit	Symbol	Relation
cubic millimetre	mm³	\( \large 1\ \text{mm}^3 = 0.000\,000\,001\ \text{m}^3 \)
cubic centimetre	cm³	\( \large 1\ \text{cm}^3 = 0.000\,001\ \text{m}^3 \)
cubic decimetre	dm³	\( \large 1\ \text{dm}^3 = 0.001\ \text{m}^3 \)
cubic metre	m³	\( \large 1\ \text{m}^3 = 1\ \text{m}^3 \)
cubic kilometre	km³	\( \large 1\ \text{km}^3 = 1\,000\,000\,000\ \text{m}^3 \)

 

 

What does cubic metre mean?

A cubic metre is the volume of a cube where each edge is one metre long.

 

$$ \large 1\ \text{m}^3 = 1\ \text{m} \cdot 1\ \text{m} \cdot 1\ \text{m} $$

 

In the same way, a cubic centimetre \( \text{cm}^3 \) means a cube with sides of one centimetre. When converting between volume units, you must remember that the length unit appears three times — once for each dimension (length, width and height).

 

 

Conversion between volume units

The conversion between volume units follows the same idea as for length and area, but because there are three dimensions, the conversion factor must be raised to the third power. This means you multiply or divide by \( 10^3 = 1000 \) for each step instead of by 10 or 100.

 

From metres to centimetres, for example:

 

$$ \large 1\ \text{m} = 100\ \text{cm} $$

 

Therefore:

 

$$ \large 1\ \text{m}^3 = (100\ \text{cm})^3 = 100^3\ \text{cm}^3 = 1\,000\,000\ \text{cm}^3 $$

 

Here we see that the number 100 is raised to the third power because volume has three dimensions.

 

 

Volume and litre

Volume is often measured in litres, especially when dealing with liquids. However, a litre is not a fundamental SI unit, but it is closely related to the cubic metre:

 

$$ \large 1\ \text{L} = 1\ \text{dm}^3 = 10^{-3}\ \text{m}^3 $$

 

This also means that:

 

$$ \large 1\ \text{m}^3 = 1000\ \text{L} $$

 

 

Example

A box measures 40 cm, 30 cm and 25 cm. What is the volume in litres?

 

$$ \large V = 40 \cdot 30 \cdot 25 = 30\,000\ \text{cm}^3 $$

 

Since \( \large 1\ \text{cm}^3 = 0.001\ \text{L} \), we get:

 

$$ \large V = 30\,000 \cdot 0.001 = 30\ \text{L} $$

 

The box can therefore hold 30 litres.

 

 

Summary

When converting volume units:

 

• the conversion factor must be raised to the third power because there are three dimensions
• the decimal point is moved three places for each step instead of one or two
• you can easily convert between cubic metres and litres by remembering that \( \large 1\ \text{m}^3 = 1000\ \text{L} \)

 

 

In this way, the metric system shows a clear mathematical connection between length, area and volume, where the powers 1, 2 and 3 govern the conversion between units.