Volume and Capacity
Every question with a fully worked solution.
- Volume and Capacity - Video - Volume and Capacity 1 Watch
Theory
Capacity is how much a container holds. Find the volume in cubic units first, then convert: \(1~\text{cm}^3 = 1~\text{mL}\) and \(1~\text{m}^3 = 1~\text{kL} = 1000~\text{L}\). A tank \(1.5\) m by \(2\) m by \(4\) m holds \(12~\text{m}^3 = 12~\text{kL} = 12\,000~\text{L}\). This Year 8 maths page shows how.
Volume first, then the unit. Step 1: find the volume, \(V = Ah\) for a prism or \(V = \pi r^2 h\) for a cylinder. Step 2: convert it to a capacity unit, as a separate step.
The links. \(1~\text{cm}^3 = 1~\text{mL}\), \(1000~\text{cm}^3 = 1~\text{L}\), \(1~\text{m}^3 = 1000~\text{L} = 1~\text{kL}\), and \(1000~\text{kL} = 1\) megalitre (ML).
One unit first. Change every length to the same unit before multiplying. Centimetres give \(\text{cm}^3\), which suits mL and L. Metres give \(\text{m}^3\), which suits kL and ML.
Use the depth of the water, not the height of the container, when the container is not full.
Rain, soil and paint spread over an area make a very thin prism: volume \(=\) area \(\times\) depth. Put both in metres: \(25\) mm of rain is \(0.025\) m deep, \(1~\text{ha} = 10\,000~\text{m}^2\) and \(1~\text{km}^2 = 1\,000\,000~\text{m}^2\).
Working backwards. From \(V = Ah\): depth \(=\) volume \(\div\) base area, and length \(=\) volume \(\div\) cross-section area. Change a capacity back to cubic units before dividing. A wire is a long thin cylinder, so its length is its volume \(\div\) the area of its circular end.
Rates. Time \(=\) capacity \(\div\) rate, and amount \(=\) rate \(\times\) time. A rate per minute gives a time in minutes; divide by \(60\) for hours.
A box folded from a sheet. Cut a square of side \(x\) from each corner and fold up the sides: the box is \(x\) high, and each side of the base is \(2x\) shorter than the sheet.
Choose a sensible unit. mL for a dose of medicine, L for a bucket or a fish tank, kL for a rainwater tank or a backyard pool, ML for a dam or a harbour.
Volume and capacity units:
| Volume | Capacity |
|---|---|
| \(1~\text{cm}^3\) | \(1~\text{mL}\) |
| \(1000~\text{cm}^3\) | \(1~\text{L}\) |
| \(1~\text{m}^3\) | \(1000~\text{L} = 1~\text{kL}\) |
| \(1000~\text{m}^3\) | \(1000~\text{kL} = 1~\text{ML}\) |
| To find | Use |
|---|---|
| volume of water | base area \(\times\) depth of water |
| depth | volume \(\div\) base area |
| time to fill or empty | capacity \(\div\) rate |
| whole containers or whole days | divide, then round down |
Capacity of a container
- Change every length to one unit (cm for mL and L, m for kL and ML).
- Find the volume: base area \(\times\) height (or depth of the water).
- Convert: \(1~\text{cm}^3 = 1~\text{mL}\), \(1~\text{m}^3 = 1~\text{kL} = 1000~\text{L}\).
- Round only at the end, and round down for whole containers or whole days.
Working backwards
- Change the capacity to cubic units (\(1~\text{L} = 1000~\text{cm}^3\)).
- Find the base area (or cross-section area).
- Divide: depth or length \(=\) volume \(\div\) area.
| \(V\) | \(=\) | \(1.5 \times 2 \times 4\) |
| \(\) | \(=\) | \(12~\text{m}^3\) |
| \(\) | \(=\) | \(12~\text{kL}\) |
| \(\) | \(=\) | \(12\,000~\text{L}\) |
The tank holds \(12\,000\) L.
(i) How many litres does it hold when full, to the nearest litre?
(ii) Write that in kilolitres, correct to \(2\) decimal places.
The radius is \(1.6 \div 2 = 0.8\) m. Work in metres.
| \(V\) | \(=\) | \(\pi \times 0.8^2 \times 2.2\) |
| \(\) | \(=\) | \(1.408\pi\) |
| \(\) | \(=\) | \(4.4233\ldots~\text{m}^3\) |
| \(4.4233\ldots \times 1000\) | \(=\) | \(4423.3\ldots~\text{L}\) |
(i) \(4423\) L
(ii) \(1~\text{m}^3 = 1~\text{kL}\), so \(4.42\) kL.
(ii) \(20\) mm of rain falls on a catchment of \(1.5~\text{km}^2\). How many megalitres is that?
(i) \(1~\text{ha} = 10\,000~\text{m}^2\) and \(3~\text{cm} = 3 \div 100 = 0.03~\text{m}\).
(ii) \(1.5~\text{km}^2 = 1\,500\,000~\text{m}^2\) and \(20~\text{mm} = 0.02~\text{m}\).
| \(V\) | \(=\) | \(2 \times 10\,000 \times 0.03\) |
| \(\) | \(=\) | \(600~\text{m}^3 = 600~\text{kL}\) |
| \(V\) | \(=\) | \(1\,500\,000 \times 0.02\) |
| \(\) | \(=\) | \(30\,000~\text{m}^3 = 30\,000~\text{kL}\) |
(i) \(600\) kL
(ii) \(30\,000~\text{kL} \div 1000 = 30\) ML
Change the capacity to \(\text{cm}^3\): \(60~\text{L} = 60\,000~\text{cm}^3\). Then divide by the area of the circular base.
| \(A\) | \(=\) | \(\pi \times 25^2\) |
| \(\) | \(=\) | \(625\pi\) |
| \(\) | \(=\) | \(1963.49\ldots~\text{cm}^2\) |
| \(h\) | \(=\) | \(60\,000 \div 1963.49\ldots\) |
| \(\) | \(=\) | \(30.55\ldots~\text{cm}\) |
The water is about \(31\) cm deep.
(i) How many kilolitres is that?
(ii) A pump empties it at \(400\) L per minute. How many hours does that take?
| \(1.2 \times 1000\) | \(=\) | \(1200~\text{kL}\) |
| \(1200 \times 1000\) | \(=\) | \(1\,200\,000~\text{L}\) |
| \(1\,200\,000 \div 400\) | \(=\) | \(3000~\text{minutes}\) |
| \(3000 \div 60\) | \(=\) | \(50~\text{hours}\) |
(i) \(1200\) kL
(ii) \(50\) hours
(ii) \(10~\text{cm}^3\) of copper is drawn into a wire \(2\) mm across. How long is the wire, in metres, to \(2\) decimal places?
(i) The base is \(40 - 10 = 30\) cm by \(30 - 10 = 20\) cm, and the box is \(5\) cm high.
(ii) The radius is \(1\) mm \(= 0.1\) cm. Length \(=\) volume \(\div\) area of the circular end.
| \(V\) | \(=\) | \(30 \times 20 \times 5\) |
| \(\) | \(=\) | \(3000~\text{cm}^3\) |
| \(\) | \(=\) | \(3000~\text{mL} = 3~\text{L}\) |
| \(A\) | \(=\) | \(\pi \times 0.1^2\) |
| \(\) | \(=\) | \(0.0314\ldots~\text{cm}^2\) |
| \(l\) | \(=\) | \(10 \div 0.0314\ldots\) |
| \(\) | \(=\) | \(318.3\ldots~\text{cm}\) |
(i) \(3\) L
(ii) \(318.3\ldots~\text{cm} \approx 3.18\) m
Common pitfalls
Frequently asked questions
What is the difference between volume and capacity?
Volume is the space an object takes up, in cubic units. Capacity is how much a container holds, usually in mL, L, kL or ML.
How many litres are in a cubic metre?
\(1~\text{m}^3 = 1000\) L, which is \(1\) kL.
How many millilitres are in a cubic centimetre?
Exactly one: \(1~\text{cm}^3 = 1\) mL, so \(1000~\text{cm}^3 = 1\) L.
How do I find the volume of rain on a roof?
Multiply the area of the roof in square metres by the rain depth in metres. The answer in cubic metres is the same number of kilolitres.
How do I find how deep the water is?
Change the amount of water to cubic units, then divide by the area of the base of the container.