BTT Mathematics / Primary Mathematics Learning Hub / Guide 6
Measurement assigns a number to a quantity using a stated unit. Converting the unit changes the numerical description but not the physical quantity. Perimeter measures boundary length, area measures surface coverage, and volume measures three-dimensional space. These quantities can use the same side lengths while answering different questions.
A rectangle twelve metres long and eight metres wide has perimeter forty metres and area ninety-six square metres. Both answers use twelve and eight, but the operations, meanings and units are different. Writing “96 m” after multiplying the side lengths does not turn area into perimeter; the calculation itself identifies a different quantity.
This guide develops unit sense, conversion, time, perimeter, area, composite shapes and volume. It aligns with the Measurement and Geometry strand of the current MOE Primary Mathematics syllabus. Select only the sections appropriate to the learner’s current school scope.
Understand units · Convert without changing the quantity · Perimeter · Area · Volume · 24 questions · Worked answers
1. A measurement is a number together with a unit
“Five” is not yet a complete length. Five centimetres, five metres and five kilometres describe very different distances. The unit tells us what one counted step means.
To measure a pencil with centimetres, place equal centimetre units along its length without gaps or overlaps. A ruler packages those equal units into a scale. Reading the endpoint is useful only when the starting point and the unit are understood.
If an object begins at the 2 cm mark and ends at the 11 cm mark, its length is not eleven centimetres. The ruler coordinate changed from two to eleven, so the length is 11 − 2 = nine centimetres. Coordinates on a scale and measured lengths are related but not identical.
Choose a unit that matches the size of the quantity
Millimetres may be useful for the thickness of a small object, centimetres for a book, metres for a classroom and kilometres for a road journey. The same length can be stated in several units, but some descriptions are more convenient.
A classroom length of eight metres could be written as eight hundred centimetres. Both are correct. Choosing metres keeps the number manageable. A tiny object measured as 0.003 kilometres is also possible but unnecessarily difficult to interpret for an ordinary primary measurement task.
Mass, capacity and time are different quantities
Grams and kilograms measure mass. Millilitres and litres measure capacity or liquid volume in everyday school contexts. Seconds, minutes and hours measure duration. Similar numerical conversions do not make the quantities interchangeable.
One kilogram equals one thousand grams, and one litre equals one thousand millilitres. This common factor of one thousand can tempt a learner to treat every unit change as a place-value shift by three digits. Time immediately shows the danger: one hour equals sixty minutes, not one thousand.
Always identify the unit family before using a conversion. The number relationship comes from the definition of those units, not from a universal decimal-moving rule.
Estimate before measuring exactly
An estimate creates a range for checking. A classroom door is likely to be around two metres high rather than two centimetres or two kilometres. A bottle might hold hundreds of millilitres or a few litres, not thousands of litres.
Estimation is not guessing without evidence. It uses known benchmarks and the size of the object. When an exact calculation later produces a wildly different order of magnitude, the estimate gives the learner a reason to inspect the unit or arithmetic.
2. Convert the unit while preserving the quantity
Example A: Metres and centimetres
One metre equals one hundred centimetres. Therefore three metres equals three hundreds of centimetres: 3 m = 300 cm.
To convert 325 centimetres to metres and centimetres, identify three complete hundreds of centimetres and twenty-five centimetres remaining. The result is 3 m 25 cm.
The physical length has not changed. A smaller unit requires more units to describe the same length. This explains why the numerical value grows when metres are converted to centimetres.
Example B: Litres and millilitres
One litre is one thousand millilitres. Thus 2.4 litres is 2,400 millilitres. The decimal 0.4 litre equals four tenths of one thousand millilitres, or four hundred millilitres.
For learners who have not yet met decimal conversions, write 2 L 400 ml. Once decimals are familiar, show that the two notations name the same capacity.
Check direction: converting litres to the smaller millilitre unit should produce a larger numerical count. If 2.4 L becomes 0.0024 ml, the direction conflicts with the unit sizes.
Example C: Kilograms and grams
A parcel has mass 4 kg 500 g. Since 4 kg = 4,000 g, the full mass is 4,500 g, or 4.5 kg.
Subtracting masses can be handled in one common unit. For 6 kg − 2 kg 750 g, write six kilograms as 6,000 g. Then 6,000 − 2,750 = 3,250 g = 3 kg 250 g.
Alternatively, regroup one kilogram as one thousand grams and subtract in mixed units. Both methods are valid when the units remain explicit.
Example D: Time uses base sixty
Two hours thirty-five minutes equals 120 minutes plus thirty-five minutes: 155 minutes. The factor between hours and minutes is sixty.
For a duration from 9:35 a.m. to 12:20 p.m., move first to a convenient boundary. From 9:35 to 10:00 is twenty-five minutes. From 10:00 to 12:00 is two hours. From 12:00 to 12:20 is twenty minutes. Total duration: 2 hours 45 minutes.
Writing 12.20 − 9.35 as though clock notation were ordinary decimal notation does not respect the sixty-minute hour. Convert or count through time units instead.
Mixed-unit addition
Add 3 m 25 cm and 1 m 80 cm. Combine centimetres: 25 + 80 = 105 cm, which is 1 m 5 cm. Combine with the four whole metres to obtain 5 m 5 cm.
Another route converts both lengths to centimetres: 325 + 180 = 505 cm = 5 m 5 cm. A common unit often simplifies arithmetic and prevents centimetres from being combined directly with metres.
3. Perimeter measures the boundary
Perimeter is the total length around a closed two-dimensional shape. Imagine walking once around the edge without cutting across the interior. Add the lengths of the boundary segments travelled.
Example E: Rectangle perimeter
A rectangle is eight centimetres long and five centimetres wide. Opposite sides have equal lengths, so its perimeter is 8 + 5 + 8 + 5 = 26 cm.
The compact formula 2 × (8 + 5) gives the same result because it counts two lengths and two widths. A formula is useful when it is recognised as a compressed statement about the boundary.
Example F: Find a missing side from perimeter
A rectangle has perimeter thirty-four centimetres and length ten centimetres. Two lengths contribute twenty centimetres. The remaining fourteen centimetres must be the two equal widths, so each width is seven centimetres.
Using P = 2(l + w), divide the perimeter by two first: 17 = 10 + w, hence w = 7. Check all four sides: 10 + 7 + 10 + 7 = 34.
Composite perimeter needs the exposed boundary only
When rectangles are joined, internal shared edges are not part of the outside perimeter. Trace the outer boundary and label each exposed segment. If a missing segment can be inferred from aligned lengths, establish that equality before adding.
Do not find the perimeter of each small rectangle and simply add them. The shared edge would be counted twice even though it lies inside the composite shape. The boundary of the combined figure is a different object from the combined boundaries of the pieces.
Same perimeter does not mean same area
A 1 cm by 9 cm rectangle has perimeter twenty centimetres and area nine square centimetres. A 4 cm by 6 cm rectangle also has perimeter twenty centimetres but area twenty-four square centimetres.
The boundary condition alone does not determine the enclosed area. This is useful when a child assumes that a larger perimeter must always enclose a larger surface.
4. Area measures how much surface is covered
Area counts equal square units needed to cover a flat region without gaps or overlaps. A rectangle eight centimetres by five centimetres can be tiled with eight columns and five rows of one-square-centimetre tiles: forty tiles.
Example G: Rectangle and square area
The area of an 8 cm by 5 cm rectangle is 8 × 5 = 40 cm². A square with perimeter thirty-six centimetres has side length nine centimetres, so its area is 9 × 9 = 81 cm².
Square centimetres are not decorative notation. One square centimetre is a square measuring one centimetre by one centimetre. Area is two-dimensional, which is why the unit is squared.
Example H: Composite area by subtraction
An L-shaped region can be viewed as a 10 cm by 8 cm rectangle with a 4 cm by 3 cm rectangular corner removed. The outer rectangle has area eighty square centimetres. The missing rectangle has area twelve square centimetres.
The L-shaped area is 80 − 12 = 68 cm². Check that the removed rectangle lies fully inside the outer one and that the stated dimensions describe its sides. A subtraction model is valid only when the geometry matches that decomposition.
Example I: Composite area by addition
The same region may be split into non-overlapping rectangles whose dimensions are known. Find each area and add. Different decompositions should give the same total if every piece is counted exactly once.
This gives a useful verification strategy. Solve an L-shape by subtraction, then redraw the split and solve by addition. Agreement between structurally different methods is stronger evidence than repeating the same multiplication twice.
Area conversion is not the same as length conversion
One metre equals one hundred centimetres. But one square metre is a square one metre on each side, so it is 100 cm × 100 cm = 10,000 cm².
Multiplying an area in square metres by one hundred would convert only one dimension. Both dimensions scale by one hundred, so the area scales by 100 × 100.
Similarly, if every side length of a rectangle is doubled, the perimeter doubles but the area becomes four times as large. The boundary has one factor of length; the surface has two.
Example J: Mixed units in an area problem
A rectangle measures 250 cm by 80 cm. In centimetres, its area is 250 × 80 = 20,000 cm². Convert the side lengths first: 2.5 m by 0.8 m gives area 2.0 m².
Both forms agree because 2 m² equals 20,000 cm². Mixing 250 cm with 0.8 m in one multiplication without converting would produce a number whose unit meaning is unclear.
5. Volume measures three-dimensional space
A one-centimetre cube has volume one cubic centimetre. A rectangular cuboid can be filled by layers of such cubes. If one layer contains length × width cubes and there are height layers, the volume is length × width × height.
Example K: Cuboid volume
A box measures 5 cm by 4 cm by 3 cm. One layer contains 5 × 4 = 20 unit cubes. Three layers contain 20 × 3 = 60 cm³.
The cubic unit records three dimensions. The numerical product is not an area because a third length has been included. As with area, the unit helps name what the calculation represents.
Example L: Find a missing dimension
A cuboid has volume 240 cm³, length 10 cm and width 6 cm. The base area is 10 × 6 = 60 cm². Since volume = base area × height, the height is 240 ÷ 60 = 4 cm.
Check by reconstructing the volume: 10 × 6 × 4 = 240. The missing height is a length, so its unit is centimetres, not cubic centimetres.
Capacity and geometric volume
For common metric units, one cubic centimetre corresponds to one millilitre, and one thousand cubic centimetres correspond to one litre. This connects a container’s internal geometric volume with liquid capacity.
Use this relationship only when the dimensions describe the internal space or when the problem explicitly treats the container that way. Thick walls and external dimensions can matter in real objects. Mathematical models depend on which dimensions are given.
Scaling all dimensions
If all three dimensions of a cuboid double, the volume is multiplied by 2 × 2 × 2 = 8. The volume does not merely double. One factor of two appears for each independent dimension.
This is a powerful extension because it connects geometry to multiplication structure. It also provides a check on intuition: a box twice as long, twice as wide and twice as high can hold eight copies of the original box.
6. Practice: 24 questions
Keep the worked answers covered. Write the unit on every final measurement. Where conversion is needed, state the common unit before calculating.
Questions 1–8: Units and direct measurement
1. Convert 3 m to centimetres.
2. Convert 2.4 L to millilitres.
3. Convert 4 kg 500 g to grams.
4. Convert 2 h 35 min to minutes.
5. A rectangle is 8 cm long and 5 cm wide. Find its perimeter.
6. Find the area of the same 8 cm by 5 cm rectangle.
7. A square has perimeter 36 cm. Find its side length and area.
8. A rectangle has perimeter 34 cm and length 10 cm. Find its width.
Questions 9–16: Convert, combine and infer
9. Add 3 m 25 cm and 1 m 80 cm.
10. Calculate 6 kg − 2 kg 750 g.
11. Five litres of juice are shared equally into eight containers. How many millilitres go into each container?
12. Find the duration from 9:35 a.m. to 12:20 p.m.
13. An L-shape is a 10 cm by 8 cm rectangle with a 4 cm by 3 cm rectangular corner removed. Find its area.
14. Find the area of a rectangular floor 12 m by 7 m.
15. A cuboid measures 5 cm by 4 cm by 3 cm. Find its volume.
16. A cuboid has volume 240 cm³, length 10 cm and width 6 cm. Find its height.
Questions 17–24: Scale and apply
17. How many square centimetres are in 1 m²?
18. A rectangle measures 250 cm by 80 cm. Find its area in cm² and in m².
19. A rectangle is 4 cm by 7 cm. Every side length is doubled. By what factor do its perimeter and area change?
20. Are 2.4 m and 240 cm the same length? Explain.
21. A tank has capacity 18 L and currently contains 12.75 L. How much more liquid can it hold? Give the answer in litres and millilitres.
22. A runner starts at 7:48 a.m. and finishes at 9:13 a.m. Find the duration.
23. A rectangular garden is 15 m by 9 m. Find the perimeter and area.
24. The garden in question 23 is fenced at $12 per metre and covered with grass at $8 per square metre. Find the fencing cost, grass cost and total cost.
7. Worked answers
Answers 1–8
1. 300 cm. One metre is one hundred centimetres, so three metres contain three hundreds of centimetres. The physical length remains unchanged.
2. 2,400 ml. One litre is one thousand millilitres. Two litres give two thousand millilitres and 0.4 litre gives four hundred millilitres.
3. 4,500 g. Four kilograms equal four thousand grams. Add the existing five hundred grams.
4. 155 min. Two hours contain 120 minutes. Add thirty-five minutes to obtain 155.
5. 26 cm. Add the four boundary sides: 8 + 5 + 8 + 5. The unit is centimetres because perimeter is a length.
6. 40 cm². Cover the rectangle with eight columns and five rows of unit squares. Multiply 8 × 5 = 40 square centimetres.
7. Side 9 cm; area 81 cm². A square has four equal sides, so 36 ÷ 4 = 9 cm. Square the side length: 9 × 9 = 81 cm².
8. 7 cm. Half the perimeter is 17 cm, representing one length plus one width. Subtract the 10 cm length to obtain a 7 cm width.
Answers 9–16
9. 5 m 5 cm. Twenty-five plus eighty gives 105 cm, or 1 m 5 cm. Add that extra metre to the four whole metres.
10. 3 kg 250 g. Convert to grams: 6,000 − 2,750 = 3,250 g. Convert back to mixed units.
11. 625 ml. Five litres equal five thousand millilitres. Divide 5,000 by eight to obtain 625 millilitres per container.
12. 2 h 45 min. From 9:35 to 10:00 is 25 minutes, then two hours to noon and 20 more minutes to 12:20. The minutes total forty-five.
13. 68 cm². The complete 10 by 8 rectangle has area 80 cm². Subtract the missing 4 by 3 rectangle, area 12 cm².
14. 84 m². Multiply the two perpendicular dimensions: 12 × 7 = 84. The unit is square metres.
15. 60 cm³. Multiply 5 × 4 × 3. A layer contains twenty unit cubes and there are three layers.
16. 4 cm. The 10 by 6 base has area 60 cm². Divide the volume 240 cm³ by the base area to find the height: four centimetres.
Answers 17–24
17. 10,000 cm². A one-metre square is 100 cm by 100 cm. Its area is 100 × 100 = 10,000 square centimetres.
18. 20,000 cm² = 2 m². Multiply 250 × 80 = 20,000 cm². Since 10,000 cm² equals 1 m², divide by 10,000 to obtain 2 m².
19. Perimeter ×2; area ×4. Doubling both dimensions doubles every boundary length, so perimeter doubles. Area contains both scale factors, giving 2 × 2 = 4 times the original area.
20. Yes. Convert 2.4 metres to centimetres: 2.4 × 100 = 240 cm. The numerical forms differ because the units differ, but the length is the same.
21. 5.25 L = 5,250 ml. Subtract 12.75 from 18.00 to obtain 5.25 litres. Multiplying by one thousand gives 5,250 millilitres.
22. 1 h 25 min. From 7:48 to 8:00 is twelve minutes, from 8:00 to 9:00 is one hour, and from 9:00 to 9:13 is thirteen minutes. Twelve plus thirteen is twenty-five minutes.
23. Perimeter 48 m; area 135 m². Perimeter is 15 + 9 + 15 + 9 = 48 m. Area is 15 × 9 = 135 m². The same dimensions produce different quantities.
24. Fencing $576; grass $1,080; total $1,656. Multiply 48 m by $12 per metre for the boundary cost. Multiply 135 m² by $8 per square metre for coverage. Add the two costs.
8. Repair the quantity before repairing the formula
If a learner uses area when perimeter is required, ask what physical task is being described. Is material going around an edge or covering a surface? Draw or trace that part. The formula should follow the quantity, not be selected from two memorised expressions after the numbers are seen.
If unit conversions fail, ask which unit is smaller and whether more or fewer of those units are needed to describe the same amount. This direction check can expose a conversion written backwards before any multiplication is repeated.
Use unit analysis as a checking habit
Adding 3 m to 25 cm without conversion mixes different units. Multiplying 12 m by 7 m produces square metres because two length dimensions are multiplied. Dividing cubic centimetres by square centimetres leaves centimetres, which is why volume divided by base area gives a height.
Primary learners do not need formal dimensional analysis notation to benefit from this habit. Simply asking “What does this answer measure?” can reveal whether the operation and unit belong together.
Separate a diagram error from an arithmetic error
For a composite shape, first inspect whether every dimension has been attached to the correct segment. A perfectly executed multiplication cannot repair a four-centimetre label accidentally placed on a five-centimetre side.
Then check whether the pieces overlap or leave a gap. Only after the geometry is sound should calculation fluency become the main focus.
Use a second representation for verification
Convert mixed units into one unit and calculate, then compare with a regrouping method. Find an L-shaped area by subtracting a missing rectangle, then split the same shape into two smaller rectangles and add. Agreement between different structures provides useful evidence.
For time, count through hour boundaries and compare with a total-minute calculation. A second route should test the first route rather than merely recopy it.
Continue through the Primary Mathematics series
For the place-value exchanges behind unit conversion, use Place Value and Regrouping. For rates such as dollars per metre, use Ratio, Rate and Percentage. For shape properties, continue to Geometry, Angles, Symmetry and Coordinates.
The wider concept owner remains Geometry and Measurement. Return to the BTT Primary Mathematics Learning Hub for the complete learning-guide route.
Original learning guide. Curriculum reference checked 6 September 2026. Examples and questions are educational illustrations rather than official examination items.