PSLE Mathematics Learning Guide · Guide 47
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A drawing of a solid made from unit cubes shows only the visible surfaces. Some cubes can be hidden behind, below or inside the arrangement. Counting only what can be seen therefore gives the wrong total.
This guide builds one habit: reconstruct the solid by layers, rows or columns; distinguish visible faces from actual cubes; and use complete-cuboid structure when dimensions are known.
Guide 7 owns cube and cuboid volume. Guide 12 owns nets and face adjacency. This page takes a different spatial job: building a three-dimensional cube count from an isometric-style representation, including hidden and missing cubes.
The MOE Primary Mathematics syllabus updated October 2025 provides current curriculum context. All diagrams below are described in text or tables so the reasoning does not depend on an unprovided picture.
A unit cube represents one cubic unit of volume
If a cube has edge length 1 cm, its volume is 1 cm³.
A solid made of 24 such cubes has volume 24 cm³, provided there are no gaps or overlaps and every unit cube is counted once.
The cube count and the volume number can therefore match when one cube represents one cubic unit.
Layer counting is the safest starting method
Suppose a solid has three horizontal layers:
- bottom layer: 12 cubes;
- middle layer: 8 cubes;
- top layer: 3 cubes.
Total = 12 + 8 + 3 = 23 cubes.
A learner who counts only the 3 top cubes and visible side cubes may miss cubes completely covered by higher layers.
Column heights give another representation
Imagine a 2-by-3 footprint. The six vertical column heights are:
| 3 | 2 | 1 |
| 2 | 2 | 1 |
Total cubes = 3 + 2 + 1 + 2 + 2 + 1 = 11 cubes.
The footprint tells you where columns stand. The height tells you how many cubes are stacked at each footprint position.
A top cube requires support in ordinary stacked-cube problems
If a cube appears on top of another cube with no floating allowed, at least one supporting cube must exist directly below it.
Therefore a visible cube at height 3 implies cubes at heights 1 and 2 beneath the same column.
This is how hidden cubes are inferred from structure rather than guessed from shading.
A complete cuboid can be counted by dimensions
A cuboid is 4 cubes long, 3 cubes wide and 2 cubes high.
Total cubes = 4 × 3 × 2 = 24 cubes.
This is the same relationship as volume = length × width × height.
Layer view: each horizontal layer has 4×3 = 12 cubes, and there are 2 layers.
Find missing cubes by comparing with a complete cuboid
A solid would form a complete 5×4×3 cuboid if 8 missing cubes were restored.
Complete cuboid = 5×4×3 = 60 cubes.
Actual solid = 60 − 8 = 52 cubes.
This method is powerful when the missing part is easier to describe than the irregular solid itself.
A rectangular block removed from a cuboid can be counted separately
A 6×4×3 cuboid has a 2×2×1 block removed from one corner.
Original = 72 cubes.
Removed = 4 cubes.
Remaining = 68 cubes.
Do not subtract visible faces. Subtract actual unit cubes in the removed block.
A front view does not uniquely determine depth
If the front view shows column heights 2, 3 and 1, you know the maximum visible heights from the front. But unless the depth is specified, multiple different solids can share the same front silhouette.
For example, one row of columns could total 6 cubes. Two identical depth rows would total 12 cubes and show the same front heights.
A single two-dimensional view is therefore not always enough to determine a unique cube count.
A top view reveals footprint but not height by itself
A top view can show six occupied positions. That proves at least six cubes if each position contains at least one cube.
But the columns may have different heights. Without height information, the total could be more than six.
Use each view for the information it actually carries.
An isometric drawing encodes three directions
In an isometric-style cube drawing, edges commonly run in three repeated directions representing the three perpendicular dimensions of the solid.
Lines appear slanted on the page, but they can still represent horizontal or vertical directions in three-dimensional space.
Do not treat page slope as physical slope. Track the cube-grid directions consistently.
Count by rows when layers are hard to see
Suppose the bottom layer has rows of 5, 5 and 3 cubes.
Bottom layer = 13 cubes.
Second layer has rows of 4 and 2 cubes = 6.
Top layer has 2 cubes.
Total = 13 + 6 + 2 = 21 cubes.
Any systematic partition—layers, rows or columns—can work if every cube is counted exactly once.
Do not count the same cube in two organisational groups
If you count all cubes by horizontal layers, do not also add the vertical column totals. They are two descriptions of the same cubes.
Use a second representation as a check, not as an additional quantity.
Visible faces and cube count answer different questions
A single cube has 6 faces and volume 1 cubic unit.
Two cubes joined face-to-face have 10 exposed faces, not 12, because the touching faces become internal. But the volume is still 2 cubic units.
Do not use exposed-face count as a shortcut for cube count.
Main worked workshop: sixteen original unit-cube tasks
1.
Layers contain 9, 5 and 2 cubes.
Answer: 16 cubes.
2.
Column heights are 1, 3, 2, 2.
Answer: 8 cubes.
3.
Complete cuboid 4×4×3.
Answer: 48 cubes.
4.
Complete cuboid 7×2×5.
Answer: 70 cubes.
5.
A 5×5×2 cuboid has 6 cubes removed.
Answer: 44 cubes.
6.
A 6×3×4 cuboid loses a 2×1×3 rectangular block.
Answer: 72 − 6 = 66 cubes.
7.
A footprint has 5 occupied positions with column heights 3,3,2,1,1.
Answer: 10 cubes.
8.
A top cube is at height 4 in an ordinary vertical stack. Minimum cubes in that column?
Answer: 4.
9.
Bottom layer 15 cubes, second 10, third 4, top 1.
Answer: 30 cubes.
10.
A learner counts 12 visible top faces and says there are 12 cubes.
Repair: cubes below visible top cubes may be hidden. Reconstruct columns or layers.
11.
A learner sees a top view with 8 occupied squares and says total is exactly 8 cubes.
Repair: 8 is the minimum if each position has at least one cube; taller columns may exist.
12.
Front silhouette heights are 2,2,3. Does that uniquely determine total cubes?
Answer: not without depth information.
13.
One horizontal layer is a 4×3 rectangle with two corner cubes missing.
Answer: 12−2=10 cubes in that layer.
14.
Three identical layers each contain 10 cubes.
Answer: 30 cubes.
15.
A 3×3×3 cube is built from unit cubes. How many unit cubes?
Answer: 27.
16.
Four cubes form an L-shape in one layer, with another cube on top of the corner cube. Total?
Answer: 5 cubes.
Some views support a minimum but not a unique total
If a top view has seven occupied footprint positions, the minimum total is seven cubes.
If no maximum height is stated, there may be no finite maximum implied by that top view alone.
This is a useful information-boundary lesson: a diagram can constrain an answer without determining it completely.
Unit-cube counting is volume in discrete form
If each unit cube is 1 cm on every edge, then:
1 cube = 1 cm³.
32 cubes = 32 cm³.
For a complete cuboid, counting individual cubes and calculating length×width×height must agree.
Use Guide 7: Cube and Cuboid Volume when dimensions and reverse-volume relationships are the main job.
A net describes surfaces, not the number of unit cubes filling a solid
A cube net consists of six square faces. It folds into one hollow surface model of a cube shape; it does not mean the solid contains six unit cubes.
Guide 12 develops nets and face adjacency. Keep surface representation separate from volumetric filling.
Common spatial-counting errors
Error 1: count only visible cubes.
Error 2: treat visible faces as cubes.
Error 3: forget support cubes below upper cubes.
Error 4: add both layer totals and column totals, double-counting the same solid.
Error 5: assume one front or top view uniquely determines a three-dimensional solid.
Independent transfer check
- Layers contain 14, 9 and 4 cubes. Find total.
- Column heights are 4,3,2,2,1. Find total.
- A 5×4×2 cuboid is complete. Find unit cubes.
- Seven cubes are removed from the cuboid in Question 3. Find remaining.
- A 6×5×3 cuboid loses a 2×2×2 block. Find remaining.
- A top view has 9 occupied positions. What minimum number of cubes is guaranteed?
- Explain why a top view alone may not give a unique total.
- Explain why an upper cube can imply hidden cubes beneath it.
Independent-check answers
1. 27 cubes.
2. 12 cubes.
3. 40 cubes.
4. 33 cubes.
5. 90−8=82 cubes.
6. 9 cubes.
7. It reveals occupied footprint positions but not each column’s height.
8. Under ordinary stacked-cube conditions, a cube cannot float; support cubes fill lower positions in its column.
Parent and tutor guide
Start with physical cubes if available. Build a solid, then ask the learner to draw its layer counts before looking at it from another direction.
Next hide the back of the solid and ask which cubes can still be inferred. The goal is not visual guessing; it is reconstructing structure from constraints.
For a complete cuboid, compare two methods: count cubes by layers and calculate length×width×height. Agreement builds the connection between discrete cube counting and volume.
The learner’s final card
What is the footprint? What are the column heights? Which cubes are hidden? Can I count by layers or rows instead? Is this a complete cuboid with usable dimensions? Am I counting cubes, visible faces or volume?
Continue through Batch 12
Use Guide 45 and Guide 46. Continue with Guide 48: Rectangular Tank Liquid Levels.
Return to the PSLE Learning Guide Mathematics route.
Sources and boundaries
MOE Primary Mathematics syllabus, updated October 2025; SEAB 2026 PSLE formats.
All examples and solids are original teaching constructions. Isometric and view-based examples develop cumulative spatial reasoning and explicitly state when information is insufficient rather than assuming a hidden configuration.