Measuring volume
When you calculate a volume, the number is meaningless without its unit. Saying "2" is not the same as saying "2 litres" or "2 cubic metres": you need to know which units are being used to measure space.
The ladder of volume units
Volume is measured in cubic units: $1 \text{ cm}^3$ is the space taken up by a small cube with edges of 1 cm. Unlike length (where each step is $\times 10$), in volume each unit is 1 000 times the next one, because $10 \times 10 \times 10 = 1\,000$.
Convert to the specified units: a) $3\ \text{m}^3$ to cubic decimetres. b) $4\,500\ \text{dm}^3$ to cubic metres. c) $0,75\ \text{dm}^3$ to cubic centimetres.
Convert to the specified units:
a) $3\ \text{m}^3$ to cubic decimetres.
b) $4\,500\ \text{dm}^3$ to cubic metres.
c) $0,75\ \text{dm}^3$ to cubic centimetres.
Express in cubic decimetres: $3\ \text{m}^3\ 450\ \text{dm}^3\ 25\ \text{cm}^3$.
Express in cubic decimetres: $3\ \text{m}^3\ 450\ \text{dm}^3\ 25\ \text{cm}^3$.
Volume and capacity
Capacity measures how much liquid or grain fits inside something, and it is measured in litres. Volume and capacity are, deep down, the same idea: that is why they are related in a very simple way.
Complete in your notebook: a) $75\ \text{l} =$ $\text{dm}^3$ b) $2\,500\ \text{cm}^3 =$ litres c) $0,5\ \text{m}^3 =$ litres
Complete in your notebook:
a) $75\ \text{l} =$ ____ $\text{dm}^3$
b) $2\,500\ \text{cm}^3 =$ ____ litres
c) $0,5\ \text{m}^3 =$ ____ litres
During a storm, $45$ litres per square metre of rain fell. a) How many millimetres of water depth is this equivalent to? b) How many litres fell on a garden of…
During a storm, $45$ litres per square metre of rain fell.
a) How many millimetres of water depth is this equivalent to?
b) How many litres fell on a garden of $60\ \text{m}^2$?
c) How many cubic metres is that?
A tank contains $1,2\ \text{m}^3$ of water. How many $750$ ml bottles can be filled with it?
A tank contains $1,2\ \text{m}^3$ of water. How many $750$ ml bottles can be filled with it?
Compound form
Sometimes a volume is written split across several units, such as $3 \text{ m}^3\ 450 \text{ dm}^3$. This is called compound form: each part must be between 0 and 999 before moving on to the next unit.
Express in compound units: $12\,345\,678\ \text{mm}^3$.
Express in compound units: $12\,345\,678\ \text{mm}^3$.
Express in cubic metres and calculate: $2\ \text{dam}^3\ 15\ \text{m}^3 + 840\ \text{m}^3\ 250\ \text{dm}^3$. Express the result in hectolitres as well.
Express in cubic metres and calculate: $2\ \text{dam}^3\ 15\ \text{m}^3 + 840\ \text{m}^3\ 250\ \text{dm}^3$. Express the result in hectolitres as well.
Volume, capacity and mass
Water has a very useful property: 1 litre of water weighs 1 kg. In addition, when an object is submerged in water, it displaces a volume of water equal to its own volume. This is how Archimedes discovered, more than 2 000 years ago, how to measure the volume of an irregular object (and how to check whether a crown was made of pure gold!) without needing any formulae. And since 1 litre of water weighs 1 kg, you can also find that volume by weighing the displaced water.
A concrete block of $1,5\ \text{m} \times 0,8\ \text{m} \times 0,4\ \text{m}$ has a density of $2\,400\ \text{kg/m}^3$. a) Calculate its volume and its mass. b) A crane…
A concrete block of $1,5\ \text{m} \times 0,8\ \text{m} \times 0,4\ \text{m}$ has a density of $2\,400\ \text{kg/m}^3$.
a) Calculate its volume and its mass.
b) A crane can lift up to $1,5$ tonnes. Can it move the block?
c) Could it move two blocks at the same time? Justify your answer.
Legend has it that Archimedes discovered how to check if the king's crown was made of pure gold by measuring the water it displaced when submerged. In the diagram, a…
Legend has it that Archimedes discovered how to check if the king's crown was made of pure gold by measuring the water it displaced when submerged. In the diagram, a crown weighing $750$ g is submerged in a measuring cylinder containing water.
a) What is the volume of the crown?
b) Calculate its density in $\text{g/cm}^3$.
c) Given that the density of gold is $19,3\ \text{g/cm}^3$ and that of silver is $10,5\ \text{g/cm}^3$, could the crown be solid gold? Give reasons for your answer.
Back to the beginning: We said that 2 litres is not the same as 2 cubic metres: since $1\text{ m}^3 = 1\,000\text{ dm}^3 = 1\,000\text{ l}$, $2\text{ m}^3$ is equal to $2\,000\text{ l}$, exactly a thousand times more!