Maths · 6 Mensuration

6.1 Units of measure

Before you start

Multiplying and dividing by $10$ , $100$ and $1000$ , including with decimals.
The area of a rectangle is length × width, and the volume of a cuboid is length × width × height.
Writing large numbers in standard form, e.g. $1 000 000 = 10^{6}$ .

Syllabus objectives covered (1)

Use metric units of mass, length, area, volume and capacity in practical situations and convert quantities into larger or smaller units.

Part 1 of 6

The metric units on your syllabus

Every metric unit on your syllabus is built from a basic unit (the metre, the gram, the litre), either on its own or with a prefix that scales it by a power of ten. Learn the three prefixes and you can rebuild every length, mass and capacity conversion yourself.
kilo means one thousand. So $1$ km $= 1000$ m and $1$ kg $= 1000$ g.
centi means one hundredth. There are $100$ cm in $1$ m.
milli means one thousandth. There are $1000$ mm in $1$ m, and $1000$ ml in $1$ litre.
One more link follows from these. A metre holds $1000$ mm and $100$ cm, so each centimetre holds $1000 \div 100 = 10$ millimetres.
$1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm$
For mass and capacity there is one fact each.
$1 kg = 1000 g 1 litre = 1000 ml$
The syllabus lists mm, cm, m and km for length, g and kg for mass, and ml and l for capacity. The area and volume units are built from the length units, and that is where the next sections go.

Part 2 of 6

Multiply or divide? Decide from the size of the unit

A conversion never changes the amount. It changes only the size of the unit you count it in. A smaller unit needs more of them to make the same amount, so the number gets bigger.
To a smaller unit, multiply. For $2.7$ m in centimetres: each metre is $100$ cm, so $2.7 \times 100 = 270$ cm.
To a larger unit, divide. For $4500$ g in kilograms: every $1000$ g makes one kilogram, so $4500 \div 1000 = 4.5$ kg.
Test every answer against this rule before you move on. If you wrote $2.7$ m as $0.027$ cm, the number shrank when the unit got smaller. That cannot be right, so you divided when you should have multiplied.
For a conversion across two steps, go through the base unit. From km to cm: multiply by $1000$ to get metres, then by $100$ to get centimetres. That is $\times 100 000$ in one go, so $0.35$ km $= 35 000$ cm.

Part 3 of 6

Area units: square the length factor

An area unit is a square whose sides are one length unit long. $1$ cm² is the area of a square that measures $1$ cm by $1$ cm.
Now measure the same square in millimetres. Each side is $10$ mm, so the square holds $10$ rows of $10$ small squares, and each small square is $1$ mm². That is $10 \times 10 = 100$ mm², not $10$ .
$1 {cm}^{2} = 10 mm \times 10 mm = 100 {mm}^{2}$
graph diagram — not drawn yet
An enlarged square labelled 1 cm on its top edge and 1 cm on its left edge, divided into a 10 by 10 grid of small squares; one small square is shaded and labelled 1 mm², and the top edge also carries the label 10 mm.
One square centimetre holds 10 rows of 10 square millimetres: 100 mm² in all.
The same reasoning gives every area conversion. Write the side of the unit square in the new unit, then multiply that side by itself.
$1 m^{2} = 100 cm \times 100 cm = 10 000 {cm}^{2}$
For square kilometres the side is $1000$ m.
$1 {km}^{2} = 1000 m \times 1000 m = 1 000 000 m^{2} = 10^{6} m^{2}$
In the same way, $1$ m² $= 1000 \times 1000 = 1 000 000$ mm². You do not need to memorise a list. When you are not sure of a factor, write the side of the unit square in the new unit and square it.
The multiply-or-divide rule still holds. $3.2$ m² in cm²: the unit is smaller, so multiply by $10 000$ to get $32 000$ cm². $6500$ cm² in m²: the unit is larger, so divide by $10 000$ to get $0.65$ m².
The trap is to use the length factor, $100$ , for an area. $1$ m² is not $100$ cm². A square metre is $100$ cm wide and $100$ cm tall, and both directions count.

Part 4 of 6

Volume units: cube the length factor

A volume unit is a cube whose edges are one length unit long. A cube that is $1$ cm along each edge has a volume of $1$ cm³.
Measure that cube in millimetres. It is $10$ mm along each of its three edges, so it holds $10 \times 10 \times 10$ small cubes of $1$ mm³.
$1 {cm}^{3} = 10 \times 10 \times 10 {mm}^{3} = 1000 {mm}^{3}$
From metres to centimetres the length factor is $100$ , so cube it.
$1 m^{3} = 100 \times 100 \times 100 {cm}^{3} = 1 000 000 {cm}^{3} = 10^{6} {cm}^{3}$
From metres to millimetres the length factor is $1000$ , so $1$ m³ $= 1000^{3} = 10^{9}$ mm³. Numbers this large are easier to handle in standard form.
The pattern: you use the length factor once for a length, twice for an area and three times for a volume. For metres to centimetres that gives the three factors below.
$length: \times 100 area: \times 100^{2} = 10 000 volume: \times 100^{3} = 1 000 000$
Example: $0.004$ m³ in cm³. The unit is smaller, so multiply by $1 000 000$ to get $4000$ cm³.

Part 5 of 6

Volume and capacity: one amount, two sets of units

Capacity is how much a container holds, and it is measured in millilitres and litres. Volume and capacity both measure an amount of space, so you can convert one into the other.
The link is that one millilitre is exactly one cubic centimetre.
$1 ml = 1 {cm}^{3}$
A litre is $1000$ ml, so it is $1000$ cm³. A cube $10$ cm along each edge has a volume of $10 \times 10 \times 10 = 1000$ cm³, so it holds exactly $1$ litre.
$1 litre = 1000 {cm}^{3}$
For cubic metres: $1$ m³ $= 1 000 000$ cm³ $= 1 000 000$ ml. Divide by $1000$ to change millilitres into litres.
$1 m^{3} = 1000 litres$
So, from cm³ to litres, divide by $1000$ . From m³ to litres, multiply by $1000$ , because a litre is smaller than a cubic metre and you need more of them.
The trap: $1$ ml is not $1$ mm³. The milli- in millilitre means one thousandth of a litre, and one thousandth of $1000$ cm³ is $1$ cm³.

Part 6 of 6

Practical problems: one unit for every length, then calculate

In a practical question the measurements can come in mixed units: a tank $1.2$ m long but $75$ cm wide, or a garden in metres with seed in grams per square metre.
Before you multiply anything, choose one unit and convert every length into it. Then calculate the area or the volume. This way you only ever convert lengths with the simple factors, and the squared and cubed factors take care of themselves.
Example: a cuboid $1.2$ m by $75$ cm by $50$ cm. Write $1.2$ m as $120$ cm, then $120 \times 75 \times 50 = 450 000$ cm³. If you multiply $1.2 \times 75 \times 50$ instead, you get $4500$ , which is not a volume in any unit.
Work in the unit the answer asks for, or in the one that avoids awkward decimals. If the answer must be in litres, working in cm³ and dividing by $1000$ at the end is a tidy route.
When the answer counts whole objects, the situation decides the rounding. If the question asks how many can be made or filled from a fixed amount, the part left over is not enough for one more, so round down.
If the question asks how many are needed to cover or hold everything, a part-used one must still be bought, so round up.
End with a size check against the real world. A classroom floor is tens of square metres, not hundreds of thousands. A bucket holds a few litres. An answer far from what makes sense points to a conversion done the wrong way.

Worked examples

Example 1. Write (i) $3.6$ m² in cm², (ii) $45 000$ cm² in m².

Show the working

Find the factor first. $1$ m $= 100$ cm, so $1$ m² $= 100 \times 100 = 10 000$ cm².
(i) The cm² is the smaller unit, so the number gets bigger: multiply.
$3.6 \times 10 000 = 36 000 {cm}^{2}$
(ii) The m² is the larger unit, so the number gets smaller: divide.
$45 000 \div 10 000 = 4.5 m^{2}$
Check the size. A square $2$ m by $2$ m is $4$ m², and it is $200$ cm by $200$ cm $= 40 000$ cm². $45 000$ cm² is a little more than that, so $4.5$ m² is the right size. Dividing by $100$ would have given $450$ m², which is far too big.

Answer: (i) $36 000$ cm² (ii) $4.5$ m²

Example 2. A fish tank is a cuboid that measures $80$ cm by $35$ cm by $40$ cm. Calculate how many litres of water it holds when it is full. Give its volume in m³ as well.

Show the working

All three lengths are already in centimetres, so calculate the volume directly.
$80 \times 35 \times 40 = 112 000 {cm}^{3}$
Change to litres. $1$ litre $= 1000$ cm³, and a litre is the larger unit, so divide.
$112 000 \div 1000 = 112 litres$
For m³ the factor is $100^{3} = 1 000 000$ , and m³ is the larger unit, so divide.
$112 000 \div 1 000 000 = 0.112 m^{3}$
Cross-check with $1$ m³ $= 1000$ litres: $0.112 \times 1000 = 112$ litres, which agrees.

Answer: $112$ litres, which is $0.112$ m³.

Example 3. A rectangular floor measures $4.2$ m by $3$ m. It is covered with square tiles of side $30$ cm, with no gaps. Work out how many tiles are needed.

Show the working

The floor is in metres and the tiles are in centimetres. Convert the floor's lengths to centimetres before you do anything else: $4.2$ m $= 420$ cm and $3$ m $= 300$ cm.
Count how many tiles fit along each side.
$420 \div 30 = 14 300 \div 30 = 10$
Multiply the number of rows by the number in each row.
$14 \times 10 = 140$
Check by area. Floor: $4.2 \times 3 = 12.6$ m² $= 126 000$ cm². One tile: $30 \times 30 = 900$ cm². Then $126 000 \div 900 = 140$ , so the two methods agree.
See what happens if you do not convert: $12.6 \div 900 = 0.014$ , which is a nonsense number of tiles. The mixed units cause the error, not the arithmetic.

Answer: $140$ tiles.

Example 4. A bag holds $2.5$ kg of rice. One portion uses $75$ g. Work out how many full portions the bag gives, and how much rice is left over.

Show the working

Put both masses in grams, the smaller unit, so that there are no decimals: $2.5$ kg $= 2.5 \times 1000 = 2500$ g.
Divide the total by one portion.
$2500 \div 75 = 33.3 \dots$
The question asks for full portions from a fixed amount, so round down to $33$ . The part after the decimal point is not enough rice for another portion.
Find what is left over.
$2500 - 33 \times 75 = 2500 - 2475 = 25 g$

Answer: $33$ full portions, with $25$ g left over.

Where people go wrong

Mistake: Writing $1$ m² $= 100$ cm², using the length factor for an area.

Instead: $1$ m² $= 100 \times 100 = 10 000$ cm².

A square metre is $100$ cm long in both directions. The length factor applies once for each direction, so for an area it is squared.

Mistake: Writing $1$ m³ $= 1000$ cm³.

Instead: $1$ m³ $= 100^{3} = 1 000 000$ cm³.

A volume has three directions, so the length factor of $100$ is cubed. The $1000$ you may be thinking of is the number of cm³ in one litre, not in one cubic metre.

Mistake: Multiplying when changing to a larger unit, e.g. writing $350$ g as $350 000$ kg.

Instead: Divide: $350$ g $= 350 \div 1000 = 0.35$ kg.

A larger unit means fewer of them. Before you calculate, decide whether the number should grow or shrink.

Mistake: Multiplying lengths in different units, such as $2$ m $\times 40$ cm $= 80$ .

Instead: Convert first: $200$ cm $\times 40$ cm $= 8000$ cm², or $2$ m $\times 0.4$ m $= 0.8$ m².

$80$ is neither m² nor cm². A product of lengths has a unit only when every length in it is in the same unit.

Mistake: Taking $1$ ml to be $1$ mm³.

Instead: $1$ ml $= 1$ cm³, so $1$ litre $= 1000$ cm³.

The milli- in millilitre means one thousandth of a litre. One thousandth of $1000$ cm³ is $1$ cm³.

Check yourself

Write $7.5$ km in metres.
Answer
$7.5 \times 1000 = 7500$ m.
Write $2350$ g in kilograms.
Answer
$2350 \div 1000 = 2.35$ kg.
Write $85$ mm² in cm².
Answer
$1$ cm² $= 100$ mm², and cm² is the larger unit, so divide: $85 \div 100 = 0.85$ cm².
Write $250 000$ m² in km².
Answer
$1$ km² $= 1 000 000$ m², so $250 000 \div 1 000 000 = 0.25$ km².
Write $4 500 000$ cm³ in m³.
Answer
$1$ m³ $= 1 000 000$ cm³, so $4 500 000 \div 1 000 000 = 4.5$ m³.
A water tank has a volume of $1.75$ m³. How many litres does it hold?
Answer
$1$ m³ $= 1000$ litres, so $1.75 \times 1000 = 1750$ litres.
A jug holds $1.5$ litres. What is its capacity in cm³?
Answer
$1$ litre $= 1000$ cm³, so $1.5 \times 1000 = 1500$ cm³.
Explain why $1$ cm³ $= 1000$ mm³.
Answer
A cube of edge $1$ cm is $10$ mm along each of its three edges, so its volume is $10 \times 10 \times 10 = 1000$ mm³.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Write each quantity in the unit given.

(a) Write $4.25$ kg in grams. 1 mark No calculator

Mark scheme

B1 $4250$ (g)

(b) Write $0.8$ m² in cm². 1 mark No calculator

Mark scheme

B1 $8000$ (cm²)

(c) Write $2 600 000$ cm³ in m³. 1 mark No calculator

Mark scheme

B1 $2.6$ (m³)

Question 2. A water tank is a cuboid. Its inside measurements are: length $1.2$ m, width $75$ cm and height $50$ cm.

(a) Calculate the volume of the tank in cm³. 2 marks

Mark scheme

M1 $1.2$ m $= 120$ cm seen, or $120 \times 75 \times 50$ , or $1.2 \times 0.75 \times 0.5$ $(= 0.45$ m³ $)$ $\times 1 000 000$ oe
A1 $450 000$ (cm³)

(b) Write the capacity of the tank in litres. 1 mark

Mark scheme

B1 $450$ (litres); FT their (a) $\div 1000$

(c) The tank contains water to a depth of $40$ cm. All of this water is poured into buckets that each hold $7$ litres. Work out how many buckets can be filled completely. 3 marks

Mark scheme

M1 $120 \times 75 \times 40 = 360 000$ cm³, or $360$ litres, seen; or their (a) $\times \frac{40}{50}$ oe
M1 their $360 \div 7 = 51.4 \dots$ , or $360 000 \div 7000$ , with both quantities in the same unit oe
A1 $51$ cao

Question 3. A rectangular garden measures $12$ m by $7.5$ m. Grass seed is spread over the whole garden at a rate of $35$ g per square metre.

(a) Calculate the mass of seed needed. Give your answer in kilograms. 3 marks

Mark scheme

M1 $12 \times 7.5 = 90$ m² seen
M1 their $90 \times 35 = 3150$ g
A1 $3.15$ (kg) cao

(b) The seed is sold in boxes of $1.2$ kg. Work out the smallest number of boxes needed. 2 marks

Mark scheme

M1 their $3.15 \div 1.2 = 2.625$ , or $3150 \div 1200$ , with both masses in the same unit
A1 $3$ ; FT their (a), rounded up to a whole number

(c) Write the area of the garden in cm². Give your answer in standard form. 1 mark

Mark scheme

B1 $9 \times 10^{5}$ (cm²)

Question 4. A rectangular wall is $2.4$ m wide and $1.5$ m high. It is covered with square tiles of side $15$ cm, with no gaps.

(a) Show that exactly $160$ tiles are needed. 2 marks No calculator

Mark scheme

M1 both lengths in cm $(240$ and $150)$ each divided by $15$ , giving $16$ and $10$ ; or $3.6$ m² written as $36 000$ cm² and divided by $15 \times 15 = 225$ ; or working in metres, $2.4 \div 0.15$ and $1.5 \div 0.15$ , or $3.6 \div 0.0225$ oe
A1 $16 \times 10 = 160$ , or $36 000 \div 225 = 160$ , or $3.6 \div 0.0225 = 160$ , with full working; nfww

(b) Tomas works out $2.4 \times 1.5 = 3.6$ and $15 \times 15 = 225$ , and then $3.6 \div 225 = 0.016$ tiles. Explain what he has done wrong, and show how to correct it. 2 marks No calculator

Mark scheme

B1 the area of the wall is in m² but the area of a tile is in cm², so the units do not match oe
B1 $3.6$ m² $= 36 000$ cm² and $36 000 \div 225 = 160$ , or $225$ cm² $= 0.0225$ m² and $3.6 \div 0.0225 = 160$ oe

How this is examined

Turning the measurements of a container into its capacity in litres: a Calculate after the length, width and height of a cuboid tank are given, possibly in two different units, followed by a request for the capacity in litres M1 for the volume with all three lengths in the same unit, e.g. $120 \times 75 \times 50$ after writing $1.2$ m as $120$ cm; A1 for the volume, e.g. $450 000$ cm³; then a mark for the change to litres: $\div 1000$ from cm³, or $\times 1000$ from m³. Write the unit after every value so the conversion can be seen.
Putting every measurement in the same unit before you calculate: a multi-step Calculate or Work out in a practical setting, such as tiles on a floor, seed spread over a garden or portions from a bag, where the data comes in mixed units A method mark for the conversion written down (e.g. $2.5$ kg $= 2500$ g), a method mark for the calculation done in matching units, and an accuracy mark for the answer with its unit. When the answer counts objects, it must be a whole number rounded the way the situation needs: down for how many can be made, up for how many are needed.
Saying exactly what is wrong when someone multiplies or divides values in different units: an Explain after a short piece of someone else's working that mixes m² and cm² One mark for naming the mismatch in specific terms, e.g. 'the wall's area is in m² but the tile's area is in cm²'. One mark for the corrected working: the converted value and the correct answer. 'The units are wrong' alone names no mismatch.
Converting an area or a volume, where the length factor must be squared or cubed: a short Write that gives a quantity in one unit and asks for it in another, e.g. m² to cm², or cm³ to m³ B1 for the converted value only, e.g. $0.8$ m² written as $8000$ cm². With a single answer mark there is no credit for the method, so the factor must be right: $10 000$ for m² and cm², $1 000 000$ for m³ and cm³. Write $1$ m² $= 100 \times 100$ at the side to check it.

Words to know

kilo-: A prefix meaning one thousand: $1$ km $= 1000$ m, $1$ kg $= 1000$ g.
centi-: A prefix meaning one hundredth: $1$ cm is one hundredth of a metre.
milli-: A prefix meaning one thousandth: $1$ mm is one thousandth of a metre, and $1$ ml is one thousandth of a litre.
Area: The amount of flat surface a shape covers, measured in square units such as mm², cm², m² and km².
Volume: The amount of space a solid takes up, measured in cubic units such as mm³, cm³ and m³.
Capacity: How much a container can hold, measured in ml and litres. $1$ ml $= 1$ cm³ and $1$ m³ $= 1000$ litres.

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