Multi-step measurement problems | Year 9

01 / LEARN

The idea, step by step

Plan before substituting

Read what the object is for. Paint covers exposed area; water occupies volume; a filling rate links capacity to time. Draw a sketch, list known dimensions and choose one unit system. Write the sequence of calculations before substituting numbers. This helps you see whether a lid, opening or hidden face should be included.

Check the practical model

A model may ignore wall thickness, assume a constant filling rate or use a stated paint coverage. Record those assumptions. A number such as 2.3 tins may require buying 3 whole tins, while a theoretical tank capacity may need spare space. Keep the mathematical result and the practical decision separate so another person can follow your reasoning.

A common trap

Using a volume calculation to work out paint, or buying a fractional number of whole containers.

Instead: Identify the type of quantity first. Make the final practical decision after calculating the required amount.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two coats of paint

A closed rectangular display box is 3 m by 2 m by 2 m. All six faces need two coats. Paint covers 8 m² per litre per coat. How many litres are required, ignoring waste?

One coat needs area $2 (3 \times 2 + 3 \times 2 + 2 \times 2) = 32$ m².
Two coats cover $64$ m². Paint needed $= 64 / 8 = 8$ L.
The model assumes the stated coverage is achieved and ignores waste.

8 L

Example 2

Fill a cylindrical tank

A tank has internal radius 0.5 m and water depth 1.2 m. Water enters at a constant 20 L per minute. How long does filling from empty take?

$V = π ({0.5}^{2}) (1.2) = 0.3 π$ m³ $= 300 π$ L.
Time $= 300 π / 20 = 15 π$ minutes $\approx 47.1$ minutes.
Use internal dimensions and assume the rate remains constant.

Approximately 47.1 minutes

Check your understanding

You need 6.4 L of paint and it is sold only in 2 L tins. How many tins must you buy?

Show answer

4 tins.

Three tins hold only 6 L. Four tins provide 8 L.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

An open-top box is 1 m long, 0.6 m wide and 0.4 m high. Find its material area, ignoring thickness.
3 marks
A cylinder has internal radius 0.4 m and height 1 m. Find its capacity to one decimal place.
3 marks
A solid consists of a 5 cm by 4 cm by 3 cm prism with a complete cylinder of radius 1 cm and height 2 cm fixed on top. The solids do not overlap. Find the total volume.
3 marks
Each closed box is 20 cm by 10 cm by 5 cm. Find the total cardboard area for 500 boxes in square metres, ignoring joins and waste.
3 marks
A rectangular tank is 80 cm by 50 cm by 40 cm internally. How many litres does it hold when three-quarters full?
3 marks
A closed cylindrical container has radius 2 m and height 3 m. Painting costs 4 dollars per square metre. Find the cost to the nearest cent, ignoring waste.
3 marks

Read the worked solutions

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

An open-top box is 1 m long, 0.6 m wide and 0.4 m high. Find its material area, ignoring thickness.
3 marks
Base $0.6$ ; long walls $2 (1 \times 0.4) = 0.8$ ; short walls $2 (0.6 \times 0.4) = 0.48$ . Total $1.88$ m².
1.88 m²
A cylinder has internal radius 0.4 m and height 1 m. Find its capacity to one decimal place.
3 marks
$V = π ({0.4}^{2}) (1) = 0.16 π$ m³ $= 160 π$ L $\approx 502.7$ L.
502.7 L
A solid consists of a 5 cm by 4 cm by 3 cm prism with a complete cylinder of radius 1 cm and height 2 cm fixed on top. The solids do not overlap. Find the total volume.
3 marks
Prism volume $= 5 \times 4 \times 3 = 60$ cm³.
Cylinder volume $= π (1^{2}) (2) = 2 π$ cm³. Add because there is no overlap.
$60 + 2 π$ cm³
Each closed box is 20 cm by 10 cm by 5 cm. Find the total cardboard area for 500 boxes in square metres, ignoring joins and waste.
3 marks
One box: $2 (200 + 100 + 50) = 700$ cm².
$500 (700) = 350000$ cm² $= 35$ m².
35 m²
A rectangular tank is 80 cm by 50 cm by 40 cm internally. How many litres does it hold when three-quarters full?
3 marks
Full volume $= 160000$ cm³ $= 160$ L.
Three-quarters of 160 L is 120 L.
120 L
A closed cylindrical container has radius 2 m and height 3 m. Painting costs 4 dollars per square metre. Find the cost to the nearest cent, ignoring waste.
3 marks
Area $= 2 π (2^{2}) + 2 π (2) (3) = 20 π$ m².
Cost $= 4 (20 π) = 80 π \approx 251.33$ dollars.
251.33 dollars

Checking saved progress…

Ready for the topic test?

Review the chapter, then choose a practice test.

Topic review

Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Year 9 surface area and volume of right prisms and cylinders, capacity, absolute/relative/percentage measurement error, precision and practical measurement problems.

Official curriculum source ↗

Back to Surface area, volume and measurement