PSLE Mathematics Learning Guide · Guide 7
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Volume problems become much easier when a cuboid is understood as a stack of equal layers rather than as three numbers to multiply without meaning.
For a cuboid, Volume = length × width × height. But the same relationship can be reorganised as Volume = base area × height. That second form is especially powerful in Primary 6 because the missing quantity may be a height, a base area or one dimension of a face.
The working habit is: identify the solid, decide which face is serving as the base, find the base area if needed, connect base area to height, then reverse the relationship carefully when a dimension is unknown.
The examples and suggested solutions are original eduKate teaching material. They are not official PSLE questions or marking schemes. The SEAB 2026 PSLE examination-format page links the Mathematics syllabus. The MOE Primary Mathematics syllabus updated October 2025 lists Primary 6 work on cube and cuboid volume, including finding one dimension from the volume and other dimensions, finding the height from volume and base area, and finding a face area from volume and one dimension.
Volume measures three-dimensional space
Length measures one dimension. Area measures two dimensions. Volume measures three dimensions.
- Length: cm
- Area: cm²
- Volume: cm³
A 5 cm by 4 cm rectangle has area 20 cm². If that rectangle forms the base of a cuboid 3 cm high, the cuboid contains three layers of 20 cm³ each, giving 60 cm³.
This layered interpretation explains why volume = base area × height.
One relationship, several useful forms
For a cuboid:
V = l × w × h.
If base area B = l × w, then:
V = B × h.
From this:
h = V ÷ B.
B = V ÷ h.
If two dimensions are known, the third can be found by dividing the volume by the product of the known dimensions.
Example: V = 240 cm³, l = 10 cm, w = 6 cm. Base area = 60 cm². Height = 240 ÷ 60 = 4 cm.
A cube is a special cuboid
All edges of a cube have the same length. If edge length is s, then:
Volume of cube = s × s × s = s³.
If s = 5 cm, volume = 125 cm³.
If the volume is 216 cm³ and the question asks for the cube’s edge length, look for the number which, multiplied by itself three times, gives 216. Since 6 × 6 × 6 = 216, the edge is 6 cm.
At this level, keep the reasoning connected to equal edge lengths rather than treating the operation as an unexplained calculator trick.
The base can be chosen, but the matching height must be perpendicular to it
A cuboid has three pairs of opposite rectangular faces. Any face can be regarded as a base if the corresponding perpendicular dimension is used as the height.
For a 10 cm × 6 cm × 4 cm cuboid:
- Base 10 × 6 = 60 cm², height 4 cm → volume 240 cm³.
- Base 10 × 4 = 40 cm², height 6 cm → volume 240 cm³.
- Base 6 × 4 = 24 cm², height 10 cm → volume 240 cm³.
The volume does not depend on which face you name as the base, provided the correct perpendicular height is paired with it.
Find one missing dimension from volume and two known dimensions
Suppose a cuboid has volume 360 cm³, length 12 cm and width 5 cm. Find its height.
Base area = 12 × 5 = 60 cm². Height = 360 ÷ 60 = 6 cm.
Check forward: 12 × 5 × 6 = 360 cm³.
The reverse check is important because division can be executed accurately even after the wrong two dimensions have been multiplied.
Find base area directly when height is known
A cuboid has volume 540 cm³ and height 9 cm. Its base area is:
540 ÷ 9 = 60 cm².
Notice the unit. Volume cm³ divided by a height cm leaves area cm². The unit relationship supports the operation.
Find a face area from volume and the perpendicular dimension
If a cuboid has volume 720 cm³ and one perpendicular dimension is 12 cm, the area of the face perpendicular to that dimension is 720 ÷ 12 = 60 cm².
This is the same volume = face area × perpendicular distance relationship. The challenge is interpreting which face the question refers to.
Convert all three dimensions before multiplying
If a cuboid measures 2 m by 50 cm by 40 cm, the dimensions are not ready to multiply as 2 × 50 × 40 if one coherent volume unit is required.
In centimetres: 2 m = 200 cm. Volume = 200 × 50 × 40 = 400,000 cm³.
In metres: 50 cm = 0.5 m and 40 cm = 0.4 m. Volume = 2 × 0.5 × 0.4 = 0.4 m³.
The numbers differ because the cubic units differ. Guide 2 develops mixed-unit control: Make Units Agree Before You Calculate.
Volume is not surface area
Volume tells you how much three-dimensional space the solid occupies. Surface area tells you how much area covers its outer faces. The formulas and units differ.
A 4 cm cube has volume 4³ = 64 cm³. Its surface area would be 6 × 4² = 96 cm². Confusing the two produces different dimensions as well as different numbers.
Before calculating, ask: inside space or outside covering?
Main worked workshop: twelve original cube-and-cuboid problems
Problem 1: direct cuboid volume
A cuboid is 8 cm long, 5 cm wide and 3 cm high. Find its volume.
Answer: 8 × 5 × 3 = 120 cm³.
Problem 2: direct cube volume
A cube has edge length 7 cm. Find its volume.
Answer: 7 × 7 × 7 = 343 cm³.
Problem 3: missing height
A cuboid has volume 480 cm³ and base dimensions 12 cm by 8 cm. Find its height.
Reasoning: Base area = 96 cm². Height = 480 ÷ 96 = 5 cm.
Problem 4: missing width
A cuboid has volume 630 cm³, length 14 cm and height 5 cm. Find its width.
Reasoning: 14 × 5 = 70. Width = 630 ÷ 70 = 9 cm.
Problem 5: find base area
A cuboid has volume 864 cm³ and height 12 cm. Find its base area.
Answer: 864 ÷ 12 = 72 cm².
Problem 6: find cube edge
A cube has volume 512 cm³. Find its edge length.
Reasoning: 8 × 8 × 8 = 512, so edge = 8 cm.
Problem 7: mixed units
A cuboid measures 1.5 m by 80 cm by 50 cm. Find its volume in cubic metres.
Reasoning: 80 cm = 0.8 m; 50 cm = 0.5 m. Volume = 1.5 × 0.8 × 0.5 = 0.6 m³.
Problem 8: compare two cuboids
Cuboid A is 10 × 6 × 4 cm. Cuboid B is 8 × 5 × 6 cm. Compare their volumes.
Reasoning: A = 240 cm³. B = 240 cm³. They have equal volume even though their dimensions differ.
Problem 9: same base, different height
Two cuboids have the same base area 35 cm². Their heights are 4 cm and 9 cm. Find the difference in volume.
Reasoning: Volume difference = base area × height difference = 35 × 5 = 175 cm³.
Problem 10: same height, different base area
Two cuboids have height 6 cm. Their base areas are 24 cm² and 39 cm². Find the difference in volume.
Reasoning: Difference in base area = 15 cm². Volume difference = 15 × 6 = 90 cm³.
Problem 11: face area from volume
A cuboid has volume 900 cm³. The distance between one pair of opposite faces is 15 cm. Find the area of either of those faces.
Reasoning: Face area = 900 ÷ 15 = 60 cm².
Problem 12: scale a cube
A cube’s edge length doubles from 3 cm to 6 cm. By what factor does volume change?
Reasoning: Edge scale factor = 2. Volume scale factor = 2³ = 8. Check: 27 cm³ to 216 cm³.
Use layers to understand missing height
If the base area is 24 cm² and the volume is 120 cm³, imagine stacking equal 24 cm³ layers of height 1 cm. Five such layers produce 120 cm³, so the height is 5 cm.
This model makes division meaningful: 120 ÷ 24 asks how many base-area layers fit into the total volume.
Audit each dimension before multiplying
Write a short dimension line:
length ___ × width ___ × height ___ = volume ___
If the same length has been used twice accidentally, the line makes it visible. If a face area has already been calculated, write base area ___ × height ___ instead of multiplying the two side lengths again.
Scaling three dimensions behaves differently from scaling one
If all edge lengths of a cuboid are doubled, volume becomes eight times as large because 2 × 2 × 2 = 8.
If only the height doubles while the base stays unchanged, volume only doubles.
If length and width both double while height stays fixed, base area becomes four times as large and volume becomes four times as large.
These scaling checks help identify whether the learner has changed one, two or three dimensions.
Independent transfer check: storage blocks
Storage Block A is a cuboid with base 12 cm by 10 cm and height 8 cm. Storage Block B has the same base area as A but a height of 15 cm. Storage Block C is a cube with volume 1000 cm³.
- Find the base area of A.
- Find the volume of A.
- Find the volume of B.
- How much greater is B’s volume than A’s?
- Find the edge length of C.
- If A’s height were unknown but its volume were 960 cm³, show how the height could be recovered.
- Explain why the base area is measured in cm² but volume is measured in cm³.
- If every dimension of A were doubled, by what factor would its volume change?
Independent-check answers
1. 12 × 10 = 120 cm².
2. 120 × 8 = 960 cm³.
3. 120 × 15 = 1800 cm³.
4. 1800 − 960 = 840 cm³.
5. 10 cm, because 10 × 10 × 10 = 1000.
6. Height = 960 ÷ 120 = 8 cm.
7. Base area uses two perpendicular length dimensions; volume includes an additional perpendicular height, giving a third dimension.
8. Eight times.
Repair the smallest reasoning defect
Draft: Volume 360 cm³, base 12 cm by 5 cm, height = 360 ÷ 12.
Problem: Only one base dimension was used.
Repair: Base area = 12 × 5 = 60; height = 360 ÷ 60.
Draft: Base area = 72 cm³.
Problem: Area was given cubic units.
Repair: 72 cm².
Draft: Cube volume 216 cm³ means edge = 216 ÷ 3.
Problem: Volume is the product of three equal edge lengths, not three times the edge.
Repair: Find s such that s × s × s = 216; s = 6.
Draft: Multiply 2 m × 40 cm × 30 cm directly and attach cm³.
Problem: Length units were mixed.
Repair: Convert all dimensions to one length unit first.
Parent and tutor guide: make the base visible
When a learner forgets what to divide by, ask them to shade one face as the base and write its area. Then ask how many such one-unit-high layers would build the volume.
If a cube problem is confused with a cuboid, ask which edge lengths are guaranteed equal. If a dimension is missing, reconstruct the whole volume relationship before performing division.
Use unit questions as a reasoning prompt: “If volume cm³ is divided by height cm, what kind of quantity should remain?” The answer cm² points toward base area.
What progress looks like
Early progress appears when the learner writes base area before finding height. Stronger progress appears when the learner can choose a different base face and still obtain the same volume.
Independent control appears when the learner checks a missing dimension by multiplying all recovered dimensions back to the original volume.
The learner’s final card
What solid is this? Which face is the base? What is its area? Which dimension is perpendicular to that face? Am I finding volume, base area or one missing length? Do the units match the dimension of the answer?
Continue through the PSLE Mathematics Learning Guide
Use Guide 5: Circle Area and Circumference, Guide 6: Composite Area and Perimeter, and continue with Guide 8: Find Unknown Angles by Naming the Geometric Rule First.
Return to the PSLE Learning Guide.
Sources and boundaries
Official references: SEAB 2026 PSLE formats and MOE Primary Mathematics syllabus, updated October 2025.
Teaching boundary: All examples and suggested solutions are original eduKate teaching material. Equivalent valid methods may exist.