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International Baccalaureate · PYP 5

Volume and Capacity

⌚ ~74 min ▢ 7 blocks ✓ 37 exercises ➣ Prerequisites : Operations with Decimals, Area, 3D Shapes and Nets, Volume and Capacity
I Units of Volume from mm³ to m³
Definition — Units of Volume
  • Cubic Millimetre \(\left(\mathrm{mm}^3\right)\): the volume of a cube with edges 1 mm long. This is about the size of a tiny grain of sand.
    FigureFigure
  • Cubic Centimetre \(\left(\mathrm{cm}^3\right)\): the volume of a cube with edges 1 cm long. This is about the size of a small die.
    Figure Figure
  • Cubic Metre \(\left(\mathrm{m}^3\right)\): the volume of a cube with edges 1 m long. Three washing machines together take up about 1 cubic metre.
    FigureFigure
Skills to practice
  • Choosing Units for Volume
II Conversion of Volume Units

Here is how volume units are related. Consider a cube with a volume of 1 \(\mathrm{cm}^3\). Since \(1 \,\text{cm} = 10 \,\text{mm}\), each side of this cube is \(10 \,\text{mm}\) long.
Figure
The volume of this cube is \(10 \,\text{mm} \times 10 \,\text{mm} \times 10 \,\text{mm}\).
The bottom layer has \(10 \times 10 = 100\) small cubes. Since the height is \(10 \,\text{mm}\), there are 10 layers.
Therefore, the total number of \(1 \,\mathrm{mm}^3\) cubes is \(100 \times 10 = 1\,000\).
$$\begin{aligned} 1 \, \text{cm}^3 &= 1 \, \text{cm} \times 1 \, \text{cm} \times 1 \, \text{cm} \\ &= 10 \, \text{mm} \times 10 \, \text{mm} \times 10 \, \text{mm} \quad (1 \, \text{cm} = 10 \, \text{mm}) \\ &= 1\,000 \, \text{mm}^3 \end{aligned}$$ So \(1 \,\text{cm}^3\) is the same as \(1\,000 \,\text{mm}^3\). This shows that when converting volume units, the conversion factor is cubed (for example, \(10\) becomes \(10^3 = 1\,000\)). The same reasoning works for the cubic metre: \(1 \,\text{m} = 100 \,\text{cm}\), so a cube of edge \(1 \,\text{m}\) holds \(100 \times 100 \times 100\) cubes of \(1 \,\text{cm}^3\), and \(1 \,\text{m}^3 = 1\,000\,000 \,\text{cm}^3\).

Proposition — Conversion of Volume Units
Because we multiply three lengths to get a volume, the conversion factors are cubed.
  • \(1 \, \text{cm} = 10 \, \text{mm}\), so \(1 \, \text{cm}^3 = (10 \times 10 \times 10) \, \text{mm}^3 = \mathbf{1\,000} \, \text{mm}^3\)
  • \(1 \, \text{m} = 100 \, \text{cm}\), so \(1 \, \text{m}^3 = (100 \times 100 \times 100) \, \text{cm}^3 = \mathbf{1\,000\,000} \, \text{cm}^3\)
Method — Converting Using Multiplication or Division
  • Use multiplication to go from a larger unit to a smaller one (like cubic metres to cubic centimetres).
  • Use division to go from a smaller unit to a larger one (like cubic centimetres to cubic metres).
Figure
Method — Converting Using the Cubed Factor
To convert a volume, we start from the factor between the lengths, then cube it.
Figure
  1. Find the volume factor between the two units: the length factor, cubed.
  2. Multiply by this factor to go to a smaller unit; divide by it to go to a larger unit.
  3. Move the decimal point: multiplying by \(1\,000\) or \(1\,000\,000\) moves it \(3\) or \(6\) places to the right; dividing moves it to the left. Fill any empty places with zeros.
We convert \(10.5 \, \text{m}^3\) to \(\text{cm}^3\), then \(7\,200\,000 \, \text{cm}^3\) to \(\text{m}^3\): $$\begin{aligned} 10.5 \, \text{m}^3 &= 10.5 \times 1\,000\,000 \, \text{cm}^3 &&\text{(}1 \, \text{m}^3 = 1\,000\,000 \, \text{cm}^3\text{)}\\ &= 10\,500\,000 \, \text{cm}^3 &&\text{(the point moves 6 places to the right)} \end{aligned}$$ $$\begin{aligned} 7\,200\,000 \, \text{cm}^3 &= (7\,200\,000 \div 1\,000\,000) \, \text{m}^3 &&\text{(}1\,000\,000 \, \text{cm}^3 = 1 \, \text{m}^3\text{)}\\ &= 7.2 \, \text{m}^3 &&\text{(the point moves 6 places to the left)} \end{aligned}$$ Check: towards a smaller unit the number gets larger (\(10.5\) becomes \(10\,500\,000\)); towards a larger unit it gets smaller (\(7\,200\,000\) becomes \(7.2\)).
Skills to practice
  • Converting Volume Units
III Litres and Millilitres with Decimals
Method — Converting Litres and Millilitres
Since \(1~\mathrm{L} = 1\,000~\mathrm{mL}\):
  • From litres to millilitres, each litre becomes \(1\,000~\mathrm{mL}\): \(3~\mathrm{L} = 3\,000~\mathrm{mL}\), and \(2~\mathrm{L}~350~\mathrm{mL} = 2\,000~\mathrm{mL} + 350~\mathrm{mL} = 2\,350~\mathrm{mL}\).
  • From millilitres to litres, group the millilitres by thousands: each thousand makes \(1~\mathrm{L}\). \(5\,000~\mathrm{mL} = 5~\mathrm{L}\), and \(1\,750~\mathrm{mL} = 1\,000~\mathrm{mL} + 750~\mathrm{mL} = 1~\mathrm{L}~750~\mathrm{mL}\).
  • Check: the millilitres left after the litres are always fewer than \(1\,000\).
Example — Litres and Millilitres with Decimals
  1. A bottle of water holds \(1.5~\mathrm{L}\). Write this capacity in millilitres.
  2. A carton of juice holds \(750~\mathrm{mL}\). Write this capacity in litres.
  3. A jug holds \(2~\mathrm{L}~300~\mathrm{mL}\). Write this capacity in litres, as a decimal.

Since \(1~\mathrm{L} = 1\,000~\mathrm{mL}\), one millilitre is a thousandth of a litre: \(1~\mathrm{mL} = 0.001~\mathrm{L}\). From litres to millilitres, multiply by \(1\,000\); from millilitres to litres, divide by \(1\,000\).
Figure
  1. From litres to millilitres: $$\begin{aligned} 1.5~\mathrm{L} &= 1.5 \times 1\,000~\mathrm{mL} &&\text{(}1~\mathrm{L} = 1\,000~\mathrm{mL}\text{)}\\ &= \textcolor{colorprop}{1\,500~\mathrm{mL}} \end{aligned}$$ Multiplying by \(1\,000\) moves the decimal point \(3\) places to the right: \(1.5\) becomes \(1\,500\). Check: \(1.5~\mathrm{L}\) is one litre and a half, and \(1\,000~\mathrm{mL} + 500~\mathrm{mL} = 1\,500~\mathrm{mL}\).
  2. From millilitres to litres: $$\begin{aligned} 750~\mathrm{mL} &= (750 \div 1\,000)~\mathrm{L} &&\text{(}1~\mathrm{mL}\text{ is a thousandth of a litre)}\\ &= \textcolor{colorprop}{0.75~\mathrm{L}} \end{aligned}$$ Dividing by \(1\,000\) moves the decimal point \(3\) places to the left: \(750\) becomes \(0.750\), that is \(0.75\). Check: \(750~\mathrm{mL}\) is less than \(1\,000~\mathrm{mL}\), and \(0.75~\mathrm{L}\) is less than \(1~\mathrm{L}\).
  3. The whole litres stay whole; the millilitres that do not make a whole litre (here \(300\), fewer than \(1\,000\)) become the decimal part: $$\begin{aligned} 2~\mathrm{L}~300~\mathrm{mL} &= 2~\mathrm{L} + (300 \div 1\,000)~\mathrm{L} &&\text{(}1~\mathrm{mL}\text{ is a thousandth of a litre)}\\ &= 2~\mathrm{L} + 0.3~\mathrm{L} &&\text{(the point moves 3 places left)}\\ &= \textcolor{colorprop}{2.3~\mathrm{L}} \end{aligned}$$ Careful with the zeros: \(30~\mathrm{mL} = 0.03~\mathrm{L}\), so \(2~\mathrm{L}~30~\mathrm{mL} = 2.03~\mathrm{L}\), not \(2.3~\mathrm{L}\).

Skills to practice
  • Deciding Which Container Holds More
  • Converting Litres and Millilitres with Decimals
  • Reading a Measuring Jug in Litres
IV Capacity and Volume

So far, volume has measured the space an object takes up, in cubic units like \(\text{cm}^3\) and \(\text{m}^3\). When we measure how much a container can hold — especially a liquid like water, milk, or juice — we talk about its capacity, and we use the litre (L).
Capacity and volume measure the same kind of thing, so their units match up exactly: $$ 1 \,\text{mL} = 1 \,\text{cm}^3, \qquad 1 \,\text{L} = 1\,000 \,\text{cm}^3, \qquad 1\,000 \,\text{L} = 1 \,\text{m}^3 . $$ Using litres and millilitres makes liquid amounts easier to compare and understand.
A box measures \(10\) cm on each side, on the inside. It is filled with water to the top: how many litres of water does it hold?

Proposition — Volume of a Rectangular Prism
The volume of a rectangular prism (also called a cuboid) is found by multiplying its length, width, and height: $$\text{Volume} = \textcolor{colordef}{\text{length}} \times \textcolor{colorprop}{\text{width}} \times \textcolor{olive}{\text{height}}$$ $$V = \textcolor{colordef}{l} \times \textcolor{colorprop}{w} \times \textcolor{olive}{h}$$
Figure
Example — From Cubic Centimetres to Litres
Leo has a cube-shaped box. Measured on the inside, it is \(10~\mathrm{cm}\) long, \(10~\mathrm{cm}\) wide and \(10~\mathrm{cm}\) high.
Figure
  1. Find the volume of the box in cubic centimetres.
  2. Leo fills the box to the top with water. How much water does it hold, in millilitres, then in litres?
  3. A water tank has a volume of \(1~\mathrm{m}^3\). How many litres of water does it hold?

  1. Inside, the box is a rectangular prism whose three edges measure \(10~\mathrm{cm}\): $$\begin{aligned} V &= \text{length} \times \text{width} \times \text{height} &&\text{(volume of a rectangular prism)}\\ &= 10~\mathrm{cm} \times 10~\mathrm{cm} \times 10~\mathrm{cm}\\ &= 100 \times 10~\mathrm{cm}^3\\ &= \textcolor{colorprop}{1\,000~\mathrm{cm}^3} \end{aligned}$$
  2. One millilitre of water fills exactly one cubic centimetre, so each of the \(1\,000\) cubic centimetres of the box holds \(1~\mathrm{mL}\): $$\begin{aligned} 1\,000~\mathrm{cm}^3 &= 1\,000~\mathrm{mL} &&\text{(}1~\mathrm{cm}^3 = 1~\mathrm{mL}\text{)}\\ &= (1\,000 \div 1\,000)~\mathrm{L} &&\text{(}1\,000~\mathrm{mL} = 1~\mathrm{L}\text{)}\\ &= \textcolor{colorprop}{1~\mathrm{L}} \end{aligned}$$ The box holds \(1\,000~\mathrm{mL}\), that is \(1~\mathrm{L}\). Its edges measure \(10~\mathrm{cm} = 1~\mathrm{dm}\): it is the cubic decimetre. So one litre is one cubic decimetre: $$1~\mathrm{L} = 1~\mathrm{dm}^3 = 1\,000~\mathrm{cm}^3.$$
  3. A cubic metre is a cube of edge \(1~\mathrm{m}\), that is \(100~\mathrm{cm}\), and each of its cubic centimetres holds one millilitre: $$\begin{aligned} 1~\mathrm{m}^3 &= (100 \times 100 \times 100)~\mathrm{cm}^3 &&\text{(}1~\mathrm{m} = 100~\mathrm{cm}\text{)}\\ &= 1\,000\,000~\mathrm{cm}^3 &&\text{(conversion of volume units)}\\ &= 1\,000\,000~\mathrm{mL} &&\text{(}1~\mathrm{cm}^3 = 1~\mathrm{mL}\text{)}\\ &= (1\,000\,000 \div 1\,000)~\mathrm{L} &&\text{(}1\,000~\mathrm{mL} = 1~\mathrm{L}\text{)}\\ &= \textcolor{colorprop}{1\,000~\mathrm{L}} \end{aligned}$$ The tank holds \(1\,000~\mathrm{L}\) of water: \(1~\mathrm{m}^3 = 1\,000~\mathrm{L}\).

Skills to practice
  • Converting Between Units of Volume and Capacity
  • Filling a Box with Water
V Review & Quizz
Skills to practice
  • Quiz