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\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
International Baccalaureate · PYP 4

Volume and Capacity

⌚ ~66 min ▢ 6 blocks ✓ 28 exercises ➣ Prerequisites : Decimals, Area, Subtraction, Multiplication, Division, 3D Shapes, Capacity and Volume
I Units of Volume

When we measure volume, it is important to use standard units so that everyone gets the same measurement. Non-standard units, like different-sized building blocks, can give different answers.
For volume, we use standard units like the cubic centimetre, written \(\text{cm}^3\), and the cubic metre, written \(\text{m}^3\).

Definition — Units of Volume
  • Cubic centimetre \(\left(\mathrm{cm}^3\right)\): The volume of a cube with sides \(1\ \text{cm}\) long. This is about the volume of a small ice cube.
    Figure\(\quad\)Figure
  • Cubic metre \(\left(\mathrm{m}^3\right)\): The volume of a cube with sides \(1\ \text{m}\) long. Three washing machines together take up about \(1\) cubic metre.
    Figure\(\quad\)Figure
Skills to practice
  • Choosing Units for Volume
  • Finding the Volume in Cubic Units
II Volume of a Rectangular Prism

Counting every little cube inside a rectangular prism (a box) gives its volume, but that is slow. Instead, imagine making the box taller one layer at a time and watching how the volume grows.
Each new layer adds the same number of cubes. By counting layer by layer, we spot a pattern and get a quick rule for volume: we can multiply the \(\textcolor{colordef}{length}\), the \(\textcolor{colorprop}{width}\), and the \(\textcolor{olive}{height}\).
  • Figure
    $$\begin{aligned} \text{Volume} &= \textcolor{colordef}{3} \times \textcolor{colorprop}{2} \times \textcolor{olive}{1} \\ &= 6 \, \text{cm}^3 \end{aligned}$$
  • Figure \(\quad\quad\) Figure
    $$\begin{aligned} \text{Volume} &= \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) + \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) \\ &= \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) \times \textcolor{olive}{2} \\ &= 12 \, \text{cm}^3 \end{aligned}$$
  • Figure \(\quad\quad\) Figure
    $$\begin{aligned} \text{Volume} &= \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) + \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) + \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) \\ &= \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) \times \textcolor{olive}{3} \\ &= 18 \, \text{cm}^3 \end{aligned}$$
  • Figure \(\quad\quad\) Figure
    $$\begin{aligned} \text{Volume} &= \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) + \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) + \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) + \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) \\ &= \left(\textcolor{colordef}{3} \times \textcolor{colorprop}{2}\right) \times \textcolor{olive}{4} \\ &= 24 \, \text{cm}^3 \end{aligned}$$
  • Figure
    $$\text{Volume} = \textcolor{colordef}{\text{length}} \times \textcolor{colorprop}{\text{width}} \times \textcolor{olive}{\text{height}}$$

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
Find the volume of this rectangular prism.
Figure

Using the formula for the volume of a rectangular prism: $$\begin{aligned} \text{Volume} &= \textcolor{colordef}{\text{length}} \times \textcolor{colorprop}{\text{width}} \times \textcolor{olive}{\text{height}}\\ &= 5\times 2 \times 3 \\ &= 10 \times 3 &&(5 \times 2 = 10)\\ &= 30 \, \text{cm}^3 \end{aligned}$$

Skills to practice
  • Finding the Volumes of Rectangular Prisms
  • Solving Problems
III Reading Capacity Scales
Definition — The Millilitre
The millilitre is a very small unit of capacity, written mL. One litre is one thousand millilitres: $$1~\mathrm{L} = 1\,000~\mathrm{mL}$$ A few benchmarks (approximate values):
a teaspoon \(5~\mathrm{mL}\)
a glass of water \(200~\mathrm{mL}\)
a can of juice \(330~\mathrm{mL}\)
a bottle of water \(1\,000~\mathrm{mL}\), that is \(1~\mathrm{L}\)
Method — Reading a Measuring Jug
On a measuring jug, the marks go up by equal amounts (for example one mark every \(100~\mathrm{mL}\)), but only some of them carry a number.
  1. Find what one step is worth: take two numbered marks that follow each other and count the steps between them. From \(200~\mathrm{mL}\) to \(400~\mathrm{mL}\), \(2\) equal steps make \(200~\mathrm{mL}\), so one step is \(100~\mathrm{mL}\).
  2. Start from the numbered mark just below the top of the liquid and add one step for each mark up to it: \(400~\mathrm{mL}\), then \(500~\mathrm{mL}\).
  3. Read at the top of the liquid, with the eyes level with it and the jug standing flat.
Figure
Example
Tom pours water into this measuring jug.
Figure
Find how many millilitres of water are in the jug.

  1. One step: from \(100~\mathrm{mL}\) to \(200~\mathrm{mL}\), \(2\) equal steps make \(100~\mathrm{mL}\), so one step is \(50~\mathrm{mL}\).
  2. The level: the numbered mark just below the water is \(300~\mathrm{mL}\), and the water is one step higher.
$$\begin{aligned} \text{water} &= 300~\mathrm{mL} + 50~\mathrm{mL} &&\text{(one step above } 300~\mathrm{mL}\text{)}\\ &= \textcolor{colorprop}{350~\mathrm{mL}} \end{aligned}$$

Skills to practice
  • Choosing a Reasonable Capacity
  • Reading Scales with Different Steps
IV Review & Quizz
Skills to practice
  • Quiz