What you'll learn
Volume is the amount of space a solid object takes up, capacity is the amount a container can hold, and mass is how heavy something is. Volume is measured in cubic units such as cubic centimetres (cm³) and cubic metres (m³); capacity is measured in millilitres (ml) and litres (l); mass is measured in grams (g) and kilograms (kg). The three are linked by one fact you must know by heart: 1 000 cm³ is exactly 1 litre. This topic covers counting cubes, the rule for the volume of a box, converting between millilitres and litres and between grams and kilograms, reading a scale, and solving the shopping and kitchen problems that the SEA paper builds out of them. Measurement carries 9 of the 40 items on the SEA Mathematics paper, and volume, capacity and mass appear in that block almost every year.
Key terms and definitions
Volume — the amount of space a solid object takes up. Measured in cubic units: cm³, m³.
Capacity — the amount a hollow container can hold when it is full. Measured in ml and litres.
Mass — the amount of matter in an object, which is what a balance or a kitchen scale measures. Measured in g and kg.
Cubic centimetre (cm³) — the space taken up by a cube that is 1 cm long, 1 cm wide and 1 cm tall. A centimetre cube from the classroom apparatus box is exactly this.
Cubic metre (m³) — the space taken up by a cube 1 m by 1 m by 1 m.
Millilitre (ml) — one thousandth of a litre. One millilitre of water fills exactly 1 cm³.
Litre (l) — 1 000 ml, and also 1 000 cm³. A large bottle of soft drink is usually 2 litres.
Gram (g) and kilogram (kg) — 1 kg = 1 000 g. A bag of rice is often 5 kg; a slice of bread is about 30 g.
Tonne (t) — 1 000 kg. Used for very heavy things such as a truckload of sand.
Cuboid — a box shape with six flat rectangular faces, like a shoebox or a crate. Its length, width and height are all needed to find its volume.
Core concepts
Volume is counting cubes
Picture a box built from centimetre cubes. It is 4 cubes long, 3 cubes wide and 2 cubes tall. The bottom layer has 4 × 3 = 12 cubes. There are 2 layers, so there are 12 × 2 = 24 cubes altogether. The volume is 24 cm³.
That is where the rule comes from, and it is worth saying properly: you are counting layers of cubes, not just multiplying because someone told you to.
Volume of a cuboid = length × width × height.
Volume of a cube = side × side × side, because all three edges are the same. A cube of side 6 cm has volume 6 × 6 × 6 = 216 cm³.
Why the unit is cubed
The answer counts cubes, so the unit must be a cubic unit. Write cm³ or m³, and say it as "cubic centimetres". A volume answer written as plain cm, or even as cm², is marked wrong even when the number is right.
The pattern is simple. Multiply two lengths and you get an area, so the unit is squared. Multiply three lengths and you get a volume, so the unit is cubed.
Capacity and the 1 000 cm³ fact
Volume and capacity describe the same idea from two sides. Volume asks how much space something takes up; capacity asks how much a container will hold. Because they are the same idea, we can swap between their units:
| This | equals this |
|---|---|
| 1 cm³ | 1 ml |
| 1 000 cm³ | 1 litre |
| 1 litre | 1 000 ml |
| 1 m³ | 1 000 litres |
So a tank whose volume is 25 000 cm³ holds 25 000 ÷ 1 000 = 25 litres. And a bucket that holds 9 litres has a volume of 9 × 1 000 = 9 000 cm³.
To go from cm³ to litres, divide by 1 000. To go from litres to cm³ or ml, multiply by 1 000. Dividing by 1 000 moves each digit three places to the right, so 25 000 becomes 25.
Mass: grams, kilograms and tonnes
Mass works in exactly the same thousands.
One kilogram is 1 000 g and one tonne is 1 000 kg. So 500 g is 0.5 kg, or half a kilogram; 250 g is 0.25 kg, a quarter of a kilogram; and 1 500 g is 1.5 kg.
Small to large means divide; large to small means multiply. 3 200 g = 3 200 ÷ 1 000 = 3.2 kg. 4.75 kg = 4.75 × 1 000 = 4 750 g.
These fractions of a kilogram show up constantly in a Trinidadian shop, where flour, sugar and channa are sold by the half or quarter kilogram.
Choosing a sensible unit
The paper often asks which unit fits an object. The rule is to choose the unit that gives a small, tidy number.
A teaspoon of vanilla essence is about 5 ml. A tin of condensed milk is about 400 g. A carton of orange juice is 1 litre. Kern's schoolbag is about 4 kg. A bag of cement is 42.5 kg. The water in a swimming pool is measured in cubic metres.
If an answer comes out as 0.000 3 kg or 45 000 000 ml, you have almost certainly chosen the wrong unit or slipped a conversion.
Reading a scale
Items often describe a scale in words: "the pointer is halfway between 2 kg and 3 kg". Work out what one small division is worth first, then count on from the labelled number. If a jug is marked 0, 250, 500, 750 and 1 000 ml, with five small spaces between each pair of labels, one small space is 250 ÷ 5 = 50 ml.
Adding and subtracting mixed measures
When a question mixes units, change everything to the smaller unit first, do the arithmetic, then change back at the end.
For example, 2 kg 400 g + 1 kg 850 g. In grams: 2 400 + 1 850 = 4 250 g. That is 4 kg 250 g, or 4.25 kg.
Worked examples
Example 1: A water tank
Anisa's father has a rectangular water tank that is 80 cm long, 50 cm wide and 40 cm deep. How many litres of water does it hold when it is full? How many litres are in it when it is three-quarters full?
Volume = length × width × height = 80 × 50 × 40.
Take it in two steps. 80 × 50 = 4 000. Then 4 000 × 40 = 160 000. So the volume is 160 000 cm³.
Change to litres by dividing by 1 000: 160 000 ÷ 1 000 = 160 litres.
Three-quarters full: one quarter is 160 ÷ 4 = 40 litres, so three quarters is 40 × 3 = 120 litres.
All three edges were already in centimetres, so no conversion was needed first. That will not always be true.
Example 2: Mangoes and a crate
Shanice buys 24 mangoes at the Debe market. Each mango has a mass of 250 g. The empty crate has a mass of 750 g. What is the total mass of the full crate, in kilograms?
Mass of the mangoes: 24 × 250 g. Split it — 24 × 200 = 4 800 and 24 × 50 = 1 200, so 4 800 + 1 200 = 6 000 g.
6 000 g = 6 000 ÷ 1 000 = 6 kg.
Now add the crate: 6 kg + 750 g. Working in grams, 6 000 + 750 = 6 750 g.
6 750 ÷ 1 000 = 6.75 kg, which is 6 kg 750 g.
A quick sense check: four mangoes make a kilogram because 4 × 250 = 1 000, so 24 mangoes make 24 ÷ 4 = 6 kg. That matches.
Example 3: Pouring juice into cups
Devon has a 2 litre bottle of mauby. He pours it into cups that each hold 150 ml. How many full cups can he pour, and how much mauby is left in the bottle?
Change to the same unit first. 2 litres = 2 × 1 000 = 2 000 ml.
Now divide: 2 000 ÷ 150.
Build it up. 150 × 10 = 1 500. 150 × 13 = 1 500 + 450 = 1 950. 150 × 14 = 2 100, which is too much.
So Devon can pour 13 full cups, and the mauby left over is 2 000 − 1 950 = 50 ml.
The answer to "how many full cups" must be a whole number. You cannot round 13.33 up to 14, because there is not enough mauby for a fourteenth full cup. Read the word full and let it decide the rounding.
Common mistakes and how to avoid them
Multiplying only two edges of a box. Answering 80 × 50 = 4 000 cm³ for the tank. Fix: a solid needs three measurements. Count them before you write the answer.
Writing the wrong unit. Giving a volume as cm or cm². Fix: three lengths multiplied means a cubed unit, every time.
Getting the conversion the wrong way round. Turning 2 litres into 0.002 ml. Fix: going to a smaller unit always makes the number bigger, so it must be a multiplication. Ask "should this number get bigger or smaller?" before you divide.
Using 100 instead of 1 000. This slips in from the length conversions, where 1 m = 100 cm. Fix: memorise the family — ml to litres, g to kg, m to km all use 1 000. Only centimetres to metres uses 100.
Mixing units in one calculation. Adding 2 kg and 750 g as "2 + 750". Fix: change both to grams first.
Rounding a "how many full containers" answer up. Fix: for full cups, full bottles, full boxes, always round down and state the remainder if asked.
Confusing mass with capacity. Fix: litres answer "how much room", kilograms answer "how heavy".
How parents can help at home
You do not need to remember any of this yourself. The kitchen and the grocery do the teaching.
Read the labels together. Every tin, bottle and packet in the cupboard states a capacity or a mass. Ask your child to sort five items from lightest to heaviest, or to say which of a 500 ml juice and a 1.5 litre water holds more.
Compare prices at the grocery. Two bags of rice, one 2 kg for TT$34 and one 5 kg for TT$80. Which is better value? Working out the price per kilogram is exactly the kind of two-step reasoning the paper rewards.
Use a measuring jug. Filling a 2 litre bottle from a 250 ml cup and counting the cups makes 1 litre = 1 000 ml something your child has felt rather than memorised.
When your child is stuck, the prompt is almost always "make both numbers the same unit first — say them all in grams" or "say them all in millilitres". If they still cannot start, ask, "is this asking how much space, how much it holds, or how heavy it is?" Naming which of the three it is usually unlocks the question.
Then ask them to say the answer back to you out loud with its unit. If the unit sounds sensible for the object, the answer is very probably right.
Exam technique for volume, capacity and mass
Decide which of the three the item is about before you calculate. The words give it away: holds, fills, pours, jug, bottle, tank means capacity; weighs, heavy, scale, balance means mass; space taken up, box, cubes, cuboid means volume.
Convert to a single unit before you do any arithmetic. Write the converted numbers down; do not try to hold them in your head.
For a box, write the three edges in a row with multiplication signs before you start, so you cannot leave one out.
Multiply in two steps rather than three at once: length × width first, then × height. That halves the chance of a slip.
Check the size of the answer. A domestic tank holding 160 litres is believable; 160 000 litres is not.
Finish with the unit the question asked for. If it says "in kilograms", an answer of 6 750 g scores nothing extra, even though it is the same mass — convert it.
Manage your time. Measurement is 9 of the 40 items, so do not spend so long on one tank that you lose easier Number marks later.
Quick revision summary
- Volume = length × width × height, in cubic units (cm³, m³).
- Cube: volume = side × side × side.
- 1 cm³ = 1 ml, and 1 000 cm³ = 1 litre.
- 1 litre = 1 000 ml; 1 kg = 1 000 g; 1 tonne = 1 000 kg.
- Small unit to large unit: divide by 1 000. Large to small: multiply by 1 000.
- 250 g = 0.25 kg, 500 g = 0.5 kg, 1 500 g = 1.5 kg.
- Change everything to one unit before adding, subtracting or multiplying.
- "How many full containers" answers round down; give the remainder if asked.
- Capacity is how much it holds; mass is how heavy it is. Always end with the unit the question asked for.