Worked example · Easy
Convert centimetres to millimetres
Express 6.48 cm in mm.
There are 10 mm in 1 cm, so multiply by 10.
\[6.48\times10=\boxed{64.8\text{ mm}}\]Chapter 39 • Revised
Units of measure
Learn how to choose the right units, convert lengths, masses and capacities, and apply squared and cubed conversion factors to area and volume. Work from simple conversions to examination-style measurement problems.
By the end of this chapter, you should be able to
- select sensible metric units and convert between millimetres, centimetres, metres and kilometres;
- convert kilograms and grams, litres, centilitres and millilitres;
- add, subtract, order and compare quantities after writing them in one unit;
- derive and use square and cubic unit conversions, including km², m², cm², mm² and m³, cm³, mm³;
- use the connections \(1\text{ ml}=1\text{ cm}^3\) and \(1\text{ litre}=1000\text{ cm}^3\);
- solve real-life questions using a cuboid's volume and check the units and scale of your answers.
1
Subtopic 1
Basic units of length, mass and capacity
Most measurements in this chapter use the metric system. A metric conversion changes how a quantity is written, not its actual size. Decide whether the answer should get numerically bigger or smaller before calculating.
Length
Use mm for very short lengths, cm for small objects, m for rooms and buildings, and km for long distances.
\[1\text{ km}=1000\text{ m},\qquad1\text{ m}=100\text{ cm}=1000\text{ mm},\qquad1\text{ cm}=10\text{ mm}\]Method: Multiply to change a larger unit into a smaller unit. Divide to change a smaller unit into a larger unit. If several lengths are added, first convert them all into the same unit.
Worked example · Easy
Convert millimetres to metres
Write 3575 mm in metres.
There are 1000 mm in 1 m; divide by 1000.
\[3575\div1000=\boxed{3.575\text{ m}}\]Worked example · Medium
Add measurements in different units
A journey covers 1.8 km, then 650 m, then another 2.35 km. Give the total in km.
Step 1: Convert the metres to kilometres.
\[650\text{ m}=0.65\text{ km}\]Step 2: Add.
\[1.8+0.65+2.35=\boxed{4.8\text{ km}}\]Worked example · Exam
Order lengths written in different units
Arrange 0.42 m, 39 cm and 415 mm from shortest to longest.
Write all lengths in millimetres.
\[0.42\text{ m}=420\text{ mm};\quad39\text{ cm}=390\text{ mm}\]Since \(390<415<420\), the correct order is:
39 cm, 415 mm, 0.42 m.
Mass
\[1\text{ kg}=1000\text{ g}\]If a problem includes whole kilograms, decimal kilograms and grams, converting everything to grams often makes the arithmetic easier. \(\tfrac12\) kg is 500 g and \(\tfrac14\) kg is 250 g.
Worked example · Easy
Convert a small mass
Express 0.075 kg in grams.
\[0.075\times1000=\boxed{75\text{ g}}\]
Worked example · Medium
Combine decimal and fractional masses
A recipe uses 1.25 kg of flour, 375 g of sugar and \(\tfrac12\) kg of oats. Find the total mass in kg.
Convert to grams and add.
\[1250+375+500=2125\text{ g}\]\[2125\div1000=\boxed{2.125\text{ kg}}\]Worked example · Exam
Mass remaining after several uses
A café begins with 4 kg of rice. It uses 850 g at lunch and 1.35 kg at dinner. Find the mass left in grams.
Use one unit throughout.
\[4\text{ kg}=4000\text{ g};\quad1.35\text{ kg}=1350\text{ g}\]\[4000-850-1350=\boxed{1800\text{ g}}\]Capacity
Capacity is how much liquid or gas a container can hold. Litres and millilitres are commonly used; centilitres also occur in the textbook exercises.
\[1\text{ litre}=1000\text{ ml},\qquad1\text{ cl}=10\text{ ml},\qquad100\text{ cl}=1\text{ litre}\]For measuring volume in cubic centimetres, the important link is \(1\text{ ml}=1\text{ cm}^3\). We develop this further in the next subtopic.
Worked example · Easy
Convert litres to millilitres
A jug holds 2.08 litres. Convert this capacity to ml.
\[2.08\times1000=\boxed{2080\text{ ml}}\]
Worked example · Medium
Centilitres and millilitres
Convert 37 cl to litres.
First change centilitres to millilitres, then to litres.
\[37\text{ cl}=370\text{ ml}=\boxed{0.37\text{ litres}}\]Worked example · Exam
Capacity remaining in a container
A container holds 5.2 litres of juice. Seven 450 ml bottles are filled. How many litres remain?
Step 1: Convert the starting capacity.
\[5.2\text{ litres}=5200\text{ ml}\]Step 2: Find the total poured out.
\[7\times450=3150\text{ ml}\]Step 3: Subtract and convert back.
\[5200-3150=2050\text{ ml}=\boxed{2.05\text{ litres}}\]Exam advice: Do not add \(1.2\text{ m}\) to \(35\text{ cm}\) without converting one of them. Always keep the unit in the final answer, and check that a conversion to a smaller unit produces a larger number.
2
Subtopic 2
Area and volume measures
The conversion factor changes when the measurement is squared or cubed. This follows from the number of dimensions:
The rule for squares and cubes
\[1\text{ m}=100\text{ cm}\]\[1\text{ m}^2=(100\text{ cm})^2=10\,000\text{ cm}^2\]\[1\text{ m}^3=(100\text{ cm})^3=1\,000\,000\text{ cm}^3\]If the length conversion is a factor of 10, the area factor is \(10^2=100\), and the volume factor is \(10^3=1000\).
Area conversions
\[1\text{ cm}^2=100\text{ mm}^2,\quad1\text{ m}^2=10\,000\text{ cm}^2\]\[1\text{ km}^2=1\,000\,000\text{ m}^2,\quad1\text{ hectare}=10\,000\text{ m}^2\]One square kilometre is 100 hectares. A hectare is a unit of area, not length.
Worked example · Easy
Square metres to square centimetres
Convert 3.7 m² to cm².
\[3.7\times10\,000=\boxed{37\,000\text{ cm}^2}\]
Worked example · Easy
Square millimetres to square centimetres
Write 8600 mm² in cm².
Because \(1\text{ cm}^2=100\text{ mm}^2\), divide by 100.
\[8600\div100=\boxed{86\text{ cm}^2}\]Worked example · Medium
Hectares and square kilometres
Convert 0.28 km² to hectares.
Use \(1\text{ km}^2=100\text{ hectares}\).
\[0.28\times100=\boxed{28\text{ hectares}}\]Worked example · Exam
Area with mixed units
A rectangular carpet is 3.6 m long and 250 cm wide. Find its area in m².
Convert the width first: \(250\text{ cm}=2.5\text{ m}\).
\[A=3.6\times2.5=\boxed{9\text{ m}^2}\]Volume and capacity conversions
\[1\text{ cm}^3=1000\text{ mm}^3,\qquad1\text{ m}^3=1\,000\,000\text{ cm}^3\]\[1\text{ ml}=1\text{ cm}^3,\quad1\text{ litre}=1000\text{ cm}^3,\quad1\text{ m}^3=1000\text{ litres}\]A cuboid has \(V=\text{length}\times\text{width}\times\text{height}\). All three lengths must be expressed in the same unit before multiplying.
Worked example · Easy
Cubic metres to cubic centimetres
Convert 0.006 m³ to cm³.
\[0.006\times1\,000\,000=\boxed{6000\text{ cm}^3}\]
Worked example · Medium
Cubic millimetres to cubic centimetres
Convert 24 500 mm³ to cm³.
One cm³ is 1000 mm³.
\[24\,500\div1000=\boxed{24.5\text{ cm}^3}\]Worked example · Medium
Capacity from cubic centimetres
A water container has a volume of 6750 cm³. Express its capacity in litres.
Since 1000 cm³ is 1 litre, divide by 1000.
\[6750\div1000=\boxed{6.75\text{ litres}}\]Worked example · Exam
Find an unknown height from capacity
A cuboid water tank is 40 cm long and 25 cm wide. It holds 18 litres when full. Find its height.
Step 1: Convert capacity to cubic centimetres.
\[18\text{ litres}=18\,000\text{ cm}^3\]Step 2: Use cuboid volume and rearrange.
\[40\times25\times h=18\,000\]\[h=\frac{18\,000}{1000}=\boxed{18\text{ cm}}\]
Worked example · Exam
Using the textbook carton diagram
The textbook carton has a base 90 mm by 57 mm and holds 1 litre. Find the height to the nearest mm.
Step 1: \(1\text{ litre}=1000\text{ cm}^3=1\,000\,000\text{ mm}^3\).
\[V=lwh\implies h=\frac{1\,000\,000}{90\times57}\]\[h\approx194.932\text{ mm}\approx\boxed{195\text{ mm}}\]This is a measurement question based on the supplied textbook diagram.
Exam advice: Before multiplying three dimensions, convert every length to the same unit. When counting how many whole small cubes physically fit inside a box, calculate how many fit along each edge; dividing volumes alone may overestimate the number.
Chapter 39 – Student exercises
Start with quick unit conversions, then attempt mixed-unit calculations and realistic multi-step problems. All exercises are newly written in examination style; they are not copied past-paper questions. Open Show Answer only when ready to check your steps.
A. Build confidence — basic conversions
1Centimetres to millimetres[2 marks] · Basic units of length, mass and capacity
Convert 8.34 cm to millimetres.
Show Answer
Multiply by 10 because 1 cm = 10 mm.
\[8.34\times10=\boxed{83.4\text{ mm}}\]2Metres to centimetres[2 marks] · Basic units of length, mass and capacity
Convert 0.675 m to cm.
Show Answer
Multiply by 100.
\[0.675\times100=\boxed{67.5\text{ cm}}\]3Millimetres to metres[2 marks] · Basic units of length, mass and capacity
Convert 7240 mm into metres.
Show Answer
Divide by 1000.
\[7240\div1000=\boxed{7.24\text{ m}}\]4Kilometres to metres[2 marks] · Basic units of length, mass and capacity
Convert 3.045 km into metres.
Show Answer
\[3.045\times1000=\boxed{3045\text{ m}}\]
5Kilograms to grams[2 marks] · Basic units of length, mass and capacity
Convert 0.038 kg into grams.
Show Answer
\[0.038\times1000=\boxed{38\text{ g}}\]
6Grams to kilograms[2 marks] · Basic units of length, mass and capacity
Convert 4650 g into kilograms.
Show Answer
\[4650\div1000=\boxed{4.65\text{ kg}}\]
7Centilitres to millilitres[2 marks] · Basic units of length, mass and capacity
Write 63 cl in ml.
Show Answer
One centilitre holds 10 ml.
\[63\times10=\boxed{630\text{ ml}}\]8Millilitres to litres[2 marks] · Basic units of length, mass and capacity
Convert 815 ml to litres.
Show Answer
\[815\div1000=\boxed{0.815\text{ litres}}\]
9Area: cm² to mm²[2 marks] · Area and volume measures
Convert 6.2 cm² into mm².
Show Answer
Square the length factor: \(10^2=100\).
\[6.2\times100=\boxed{620\text{ mm}^2}\]10Area: cm² to m²[2 marks] · Area and volume measures
Convert 18 600 cm² into m².
Show Answer
Divide by 10 000.
\[18\,600\div10\,000=\boxed{1.86\text{ m}^2}\]11Volume: cm³ to mm³[2 marks] · Area and volume measures
Convert 4.25 cm³ to mm³.
Show Answer
Cube the factor 10: \(10^3=1000\).
\[4.25\times1000=\boxed{4250\text{ mm}^3}\]12Volume: m³ to litres[2 marks] · Area and volume measures
Convert 0.35 m³ into litres.
Show Answer
One cubic metre contains 1000 litres.
\[0.35\times1000=\boxed{350\text{ litres}}\]B. Apply the skills — mixed units and quantities
13Two distances[3 marks] · Basic units of length, mass and capacity
A path is 1.65 km long. Another section measures 890 m. Find the total in km.
Show Answer
Convert 890 m to 0.89 km, then add.
\[1.65+0.89=\boxed{2.54\text{ km}}\]14Order by mass[3 marks] · Basic units of length, mass and capacity
Put 750 g, 0.82 kg, 0.705 kg and 95 g in order, lightest first.
Show Answer
Convert to grams: 750 g, 820 g, 705 g, 95 g.
95 g, 0.705 kg, 750 g, 0.82 kg.
15Remaining flour[3 marks] · Basic units of length, mass and capacity
A baker has 3.2 kg of flour and uses 875 g. How many grams remain?
Show Answer
Change 3.2 kg to 3200 g.
\[3200-875=\boxed{2325\text{ g}}\]16Drinks in a jug[3 marks] · Basic units of length, mass and capacity
A 2.4-litre jug is filled. A student pours out three 360 ml glasses. How much remains in litres?
Show Answer
Three glasses use \(3\times360=1080\) ml.
\[2400-1080=1320\text{ ml}=\boxed{1.32\text{ litres}}\]17Land area[3 marks] · Area and volume measures
Convert 2.75 hectares into square metres.
Show Answer
One hectare is 10 000 m².
\[2.75\times10\,000=\boxed{27\,500\text{ m}^2}\]18Square kilometres to hectares[3 marks] · Area and volume measures
A park occupies 0.064 km². How many hectares is this?
Show Answer
One km² is 100 hectares.
\[0.064\times100=\boxed{6.4\text{ hectares}}\]19Cubic metres to cm³[3 marks] · Area and volume measures
Change 0.018 m³ into cm³.
Show Answer
\[0.018\times1\,000\,000=\boxed{18\,000\text{ cm}^3}\]
20Cubic millimetres to cm³[3 marks] · Area and volume measures
Express 37 500 mm³ in cm³.
Show Answer
\[37\,500\div1000=\boxed{37.5\text{ cm}^3}\]
21Volume to capacity[3 marks] · Area and volume measures
A container holds 9250 cm³ of water. Find its capacity in litres.
Show Answer
1 litre = 1000 cm³.
\[9250\div1000=\boxed{9.25\text{ litres}}\]22Compare two areas[3 marks] · Area and volume measures
Which is larger: 0.36 m² or 3400 cm²? By how many cm²?
Show Answer
Convert the first area.
\[0.36\times10\,000=3600\text{ cm}^2\]\[3600-3400=\boxed{200\text{ cm}^2}\]So 0.36 m² is larger.
23Fill a rectangular container[3 marks] · Area and volume measures
A container has internal dimensions 30 cm by 20 cm by 15 cm. How many litres does it hold?
Show Answer
Find the volume, then divide by 1000.
\[30\times20\times15=9000\text{ cm}^3\]\[9000\div1000=\boxed{9\text{ litres}}\]24Mixed-unit rectangle[4 marks] · Area and volume measures
A noticeboard measures 1.8 m by 85 cm. Find its area in m².
Show Answer
Convert 85 cm to 0.85 m.
\[A=1.8\times0.85=\boxed{1.53\text{ m}^2}\]C. Exam-style challenge — multi-step applications
25Long-distance activity[4 marks] · Basic units of length, mass and capacity
A pupil cycles 4.25 km, jogs 780 m and walks 1250 m. Find the total distance in km.
Show Answer
Convert metres to kilometres: 780 m = 0.78 km; 1250 m = 1.25 km.
\[4.25+0.78+1.25=\boxed{6.28\text{ km}}\]26Food preparation[4 marks] · Basic units of length, mass and capacity
A canteen begins with 5 kg of rice. Each of four trays needs 675 g. Find the remaining mass in kilograms.
Show Answer
Convert all masses to grams.
\[4\times675=2700\text{ g}\]\[5000-2700=2300\text{ g}=\boxed{2.3\text{ kg}}\]27Bottling a mixture[4 marks] · Basic units of length, mass and capacity
A barrel contains 8.6 litres of juice. It is poured into 250 ml bottles. How many completely filled bottles are possible and how much juice is left?
Show Answer
Convert litres to ml, then divide.
\[8.6\text{ litres}=8600\text{ ml};\qquad8600\div250=34\text{ remainder }100\]34 full bottles, with 100 ml left.
28Calculate concrete required[5 marks] · Area and volume measures
A rectangular foundation is 6 m long, 2.5 m wide and 40 cm deep. Find its volume in m³ and litres.
Show Answer
Convert depth first: 40 cm = 0.4 m.
\[V=6\times2.5\times0.4=6\text{ m}^3\]\[6\times1000=\boxed{6000\text{ litres}}\]The volume is 6 m³, or 6000 litres.
29Unknown height of tank[5 marks] · Area and volume measures
A cuboid tank has a base 80 cm by 50 cm and a capacity of 120 litres. Find its height in cm.
Show Answer
Convert 120 litres to 120 000 cm³.
\[80\times50\times h=120\,000\]\[h=\frac{120\,000}{4000}=\boxed{30\text{ cm}}\]30Unit-conversion error[4 marks] · Area and volume measures
A student says, “3 m² = 300 cm² because 1 m = 100 cm.” Explain the mistake and give the correct value.
Show Answer
Area is two-dimensional, so multiply by \(100^2\), not 100.
\[3\times100^2=3\times10\,000=\boxed{30\,000\text{ cm}^2}\]Using the length factor instead of its square caused the error.
31Packing identical cubes[5 marks] · Area and volume measures
A rectangular box has internal dimensions 12 cm by 9 cm by 7.5 cm. How many whole cubes of side 1.5 cm can fit in a regular grid, with edges parallel?
Show Answer
Divide each dimension by the cube side, then multiply the whole-number counts.
\[12\div1.5=8,\quad9\div1.5=6,\quad7.5\div1.5=5\]\[8\times6\times5=\boxed{240\text{ cubes}}\]32Areas of two pitches[5 marks] · Area and volume measures
Two identical rectangular sports pitches have a combined area of 1.5 hectares. Each is 125 m long. Find the width of each pitch.
Show Answer
Convert hectares and divide equally.
\[1.5\text{ ha}=15\,000\text{ m}^2\]\[\text{Area per pitch}=15\,000\div2=7500\text{ m}^2\]\[\text{Width}=7500\div125=\boxed{60\text{ m}}\]Chapter 39 high-grade checklist
Correct unit choice
I use sensible mm, cm, m, km, g, kg, ml or litres and include units in answers.
Mixed measurements
I convert everything to the same unit before adding, subtracting or comparing.
Area and volume factors
I square length factors for area and cube them for volume.
Capacity versus volume
I know 1 ml = 1 cm³, 1 litre = 1000 cm³ and 1 m³ = 1000 litres.
Practical problems
I use a cuboid's volume and distinguish whole items that physically fit from their theoretical volume.
Reasonableness
I check whether the answer is larger or smaller when it should be, and give a suitable degree of accuracy.