Volume and capacity of prisms and cylinders: QCE Essential Mathematics Unit 3
“Calculate the volume and capacity of prisms, cylinders and composite solids, convert between cubic units and units of capacity, and solve practical problems involving containers, tanks, concrete and packaging”
The volume of any prism or cylinder is cross-section area times length: for a box and for a cylinder. Convert all lengths to the same unit first, remember 1 cm³ = 1 mL and 1 m³ = 1000 L, and round quantities sensibly for the practical context.
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What this dot point is asking
You need to calculate how much space a solid takes up (volume) and how much it holds (capacity), then use this for practical problems such as tanks, pools, concrete and packaging. Unit conversions are central.
The answer
Volume of prisms and cylinders
A prism has the same cross-section all the way through. For any prism:
where is the area of the cross-section and is the length (or height) of the prism.
| Solid | Volume |
|---|---|
| Rectangular prism (box) | |
| Triangular prism | |
| Cylinder |
For composite solids, split into simple solids and add (or subtract) their volumes.
Volume and capacity
- Volume uses cubic units: mm³, cm³, m³.
- Capacity uses mL, L and kL.
- 1 cm³ = 1 mL
- 1000 cm³ = 1 L
- 1 m³ = 1000 L = 1 kL
- 1 m³ = 1 000 000 cm³ (100 × 100 × 100)
Practical problems
- Convert all lengths to the same unit before multiplying.
- Concrete and soil are ordered in cubic metres; water in litres or kilolitres.
- Round up amounts you need to buy; round down the number of full containers you can fill.
A rectangular pool is 8 m long, 4 m wide and 1.4 m deep. It is filled to 90% of its depth. How many litres of water does it hold?
- Full volume = 8 × 4 × 1.4 = 44.8 m³.
- 90% full: 44.8 × 0.9 = 40.32 m³.
- Convert: 40.32 × 1000 = 40 320 L.
- Check: roughly 8 × 4 × 1.3 is about 42 m³, so about 42 000 L. Reasonable.
- Mixing units
- A slab 5 m by 4 m by 100 mm needs 100 mm converted to 0.1 m.
- Using the diameter instead of the radius in
- Converting cubic metres to litres by multiplying by 100
- It is 1000.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksA fish tank is 60 cm long, 30 cm wide and 40 cm high. Find its capacity in litres. How many litres are needed to fill it to 5 cm below the top?Show worked solution →
Full volume = 60 × 30 × 40 = 72 000 cm³ = 72 000 mL = 72 L.
Filled to 35 cm: 60 × 30 × 35 = 63 000 cm³ = 63 L.
Marking guide: 1 mark for 72 000 cm³, 1 mark for converting to 72 L, 1 mark for 63 L.
core4 marksA rainwater tank is a cylinder with diameter 2.2 m and height 2 m. How many litres does it hold when full? If a household uses 350 L per day, how many full days will a full tank last?Show worked solution →
Radius = 1.1 m.
Volume = m³ (to 2 decimal places).
Capacity = 7.60 × 1000 = 7600 L (7603 L before rounding).
Days = 7603 divided by 350 = 21.7, so 21 full days.
Marking guide: 1 mark for the radius, 1 mark for the volume, 1 mark for converting to litres, 1 mark for 21 days.
exam5 marksA concrete driveway is 12 m long, 3.2 m wide and 100 mm thick. Concrete is ordered in cubic metres, rounded up to the next 0.2 m³, and costs 280 dollars per cubic metre. Find the cost of the concrete and explain why the thickness must be converted.Show worked solution →
Convert the thickness: 100 mm = 0.1 m. All three dimensions must be in metres to give cubic metres.
Volume = 12 × 3.2 × 0.1 = 3.84 m³.
Rounded up to the next 0.2 m³: 4.0 m³.
Cost = 4.0 × $280 = $1120.
If the thickness were left as 100, the volume would be 3840 "m³", a thousand times too large, which is clearly unreasonable for a driveway.
Marking guide: 1 mark for converting mm to m, 1 mark for 3.84 m³, 1 mark for rounding to 4.0 m³, 1 mark for the cost, 1 mark for the explanation.