Volume and capacity | Year 11

01 / LEARN

Understand the idea.

Understand the idea

A prism has the same cross-section all the way along its length. Its volume equals the cross-sectional area multiplied by the perpendicular distance through which that section extends. A cylinder follows the same principle using a circular cross-section.

Capacity measures the space available inside a container. Use internal dimensions when wall thickness matters, and distinguish full capacity from the amount currently present. A partly filled tank contains only the relevant fraction of its internal volume.

Choose and apply a method

Find the cross-sectional area before multiplying by the prism length. For a triangular section, its height must be perpendicular to its base. Square units multiplied by length units give cubic units, which helps check that you used a volume calculation.

Convert to a capacity unit only after obtaining the volume in a consistent cubic unit. Use $1 m^{3} = 1000 L$ , and retain $π$ until the requested final rounding. Check whether a circular measurement is a radius or a diameter.

\begin{aligned} V_{prism} & = A_{cross-section} L \\ V_{cylinder} & = π r^{2} h \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Triangular prism

A triangular prism has cross-section base 6 cm, perpendicular height 4 cm and length 10 cm. Find volume.

Worked solution

1
Find the triangular cross-sectional area.
$A = \frac{1}{2} \times 6 \times 4 = 12 {cm}^{2}$
The stated 4 cm height is perpendicular to the 6 cm base.
2
Extend that cross-section along the prism.
$V = 12 \times 10 = 120 {cm}^{3}$
The same 12 square centimetre section extends through 10 cm of length.
3
Check the dimensions of the result.
${cm}^{2} \times cm = {cm}^{3}$
Volume must be reported in cubic units.

Answer

120 cm $3$ .

EXAMPLE 02

Tank capacity

A tank has internal radius 0.5 m and height 1.2 m. Find its capacity to the nearest litre.

Worked solution

1
Use the internal radius and height.
$V = π {(0.5)}^{2} (1.2) = 0.3 π m^{3}$
Internal dimensions determine the space available for liquid.
2
Convert the volume into litres.
$C = 1000 (0.3 π) = 300 π L$
Each cubic metre contains 1000 litres.
3
Round the full capacity as requested.
$300 π \approx 942.4778 ⟹ C \approx 942 L$
The first decimal digit is less than 5, so nearest-litre rounding gives 942 litres.

Answer

942 L.

03 / YOUR TURN

Try it yourself.

Check your understanding

A container has internal volume 0.35 m $3$ . Find capacity.

Show answer & reasoning

350 L.

Convert cubic metres to litres by multiplying by 1000: $0.35 \times 1000 = 350 L$ . This is the full capacity corresponding to the supplied internal volume.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find volume of a 7 cm by 5 cm by 4 cm cuboid.
2 marks
A prism cross-section has area 18 cm $2$ and length 12 cm. Find volume.
2 marks
A triangular prism has cross-section base 8 cm, perpendicular height 3 cm and length 9 cm. Find volume.
2 marks
A cylinder has diameter 6 cm and height 10 cm. Find exact volume.
2 marks
An internal tank 1.5 m by 0.8 m by 0.6 m is 75% full. Find water volume in litres.
2 marks
A cylinder has volume $200 π$ cm $3$ and radius 5 cm. Find height.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find volume of a 7 cm by 5 cm by 4 cm cuboid.
2 marks
$7 \times 5 \times 4 = 140$ .
140 cm $3$ .
A prism cross-section has area 18 cm $2$ and length 12 cm. Find volume.
2 marks
$18 \times 12 = 216$ .
216 cm $3$ .
A triangular prism has cross-section base 8 cm, perpendicular height 3 cm and length 9 cm. Find volume.
2 marks
$(\frac{1}{2} \times 8 \times 3) \times 9 = 108$ .
108 cm $3$ .
A cylinder has diameter 6 cm and height 10 cm. Find exact volume.
2 marks
Radius $= 3$ cm.
$V = π {(3)}^{2} (10) = 90 π$ .
$90 π$ cm $3$ .
An internal tank 1.5 m by 0.8 m by 0.6 m is 75% full. Find water volume in litres.
2 marks
Full volume $= 0.72$ m $3 = 720$ L.
$0.75 \times 720 = 540$ L.
540 L.
A cylinder has volume $200 π$ cm $3$ and radius 5 cm. Find height.
2 marks
$200 π = π {(5)}^{2} h = 25 π h$ .
$h = 8$ cm.
8 cm.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics measurement skills and applications; original explanatory lessons and papers.

Official curriculum source ↗

In this lesson

Understand the idea.

Understand the idea

Choose and apply a method

See the method in action.

Triangular prism

Find the triangular cross-sectional area.

Extend that cross-section along the prism.

Check the dimensions of the result.

Tank capacity

Use the internal radius and height.

Convert the volume into litres.

Round the full capacity as requested.

Try it yourself.

Check your understanding

Keep practising