Mathematics Tutor Singapore | PSLE Maths geometry and composite shapes are often difficult for a reason that a mark scheme does not show: the child has a good rectangle formula but has not decided whether the question asks for space inside a figure or distance around its boundary. I ask the student to trace the shape before using any equation.
A Primary 5 or Primary 6 learner can calculate area and perimeter correctly for a square and still struggle with an L-shaped composite figure. Hidden lengths, missing corners and internal lines make the picture unfamiliar. PSLE Mathematics tuition should help the child see what stays constant and what changes when a rectangle is cut apart, rather than prescribe a single sketch for every shape.
Here are worked examples of a corner cut, an inward notch and an enclosed hole, followed by new test questions, common mistakes and a practical tuition routine. All dimensions are invented teaching examples; they do not reproduce any official examination item.
Browse the composite-shapes lessons
- Area and perimeter are different questions
- Worked PSLE example: a rectangle with one corner cut away
- How to find lengths that are not labelled
- Why an inward notch changes the perimeter
- A hole has its own boundary
- A tutor’s four-way geometry error diagnosis
- Practice problems with complete answers
- How to plan PSLE geometry revision
- Frequently asked parent questions
- Sources and further Mathematics routes
Area and perimeter ask different questions
Area measures how much two-dimensional space belongs to a region. For a rectangle, area = length × breadth. A unit such as square centimetres (cm²) fits because the region can be tiled with one-centimetre squares.
Perimeter measures the total length around a specified boundary. Its units are ordinary centimetres (cm), not cm², because we are adding lengths rather than counting tiles.
That distinction becomes visible when a figure contains a notch or a cut-out. A notch removes area, but it can increase, decrease or leave unchanged the measured boundary depending on where the cut occurs and which boundary the question is asking about.
I often begin a tutoring session by asking children to imagine two jobs: painting the surface, and walking along its edge. Painting calls for area; tracing the edge calls for perimeter. Then we replace the story with precise units and mathematics.
- Mark the original enclosing rectangle if it helps organise the figure.
- Identify what material has been removed or added before calculating area.
- Trace only the boundary asked for; do not include internal construction lines as though they were outside edges.
- Find missing straight lengths using aligned horizontal and vertical totals.
- Write cm for perimeter and cm² for area when lengths are in centimetres.
- Estimate whether the area is smaller than the enclosing rectangle and whether the boundary answer is plausible.
Worked PSLE example: a corner is cut away
Imagine a rectangle measuring 12 cm wide and 8 cm tall. Remove a small rectangle 5 cm wide and 3 cm tall from the top-right corner. The remaining shape is an L-shaped figure. Find its area and perimeter.
For the area, begin with the large rectangle: 12 × 8 = 96 cm². The removed corner has area 5 × 3 = 15 cm². Therefore the remaining L-shape has area 96 − 15 = 81 cm².
Perimeter requires a new idea. Cutting away that upper-right corner removes five centimetres from the original top edge and three centimetres from the original right edge. But the new notch introduces an inward horizontal edge of five centimetres and an inward vertical edge of three centimetres. In this special corner-cut arrangement, the removed boundary length equals the added boundary length.
| Boundary segment | Length | Why it belongs |
|---|---|---|
| Bottom edge | 12 cm | Full original width |
| Remaining right edge | 8 − 3 = 5 cm | The top-right part has been cut away |
| Inward horizontal edge | 5 cm | New boundary exposed by the cut |
| Inward vertical edge | 3 cm | New boundary exposed by the cut |
| Remaining top edge | 12 − 5 = 7 cm | Top-left part that remains |
| Left edge | 8 cm | Full original height |
| Total perimeter | 12 + 5 + 5 + 3 + 7 + 8 = 40 cm | The boundary is traced once |
There is a second way to check the perimeter. The original rectangle has perimeter 2(12 + 8) = 40 cm. The corner cut removes 5 + 3 centimetres and introduces 5 + 3 centimetres. The total stays 40 cm. This is not a rule for every cut-out; it is a consequence of this exact shape.
Why a student might get 32 cm instead
A pupil may add only the four edges that resemble the original rectangle and omit the two inward edges. Another may count the original top and right sides in full and include the inward edges, counting some boundary portions that no longer exist. I ask the student to run a finger along the actual outer edge, naming each segment exactly once.
A good written solution uses short labels rather than a long explanation of what the diagram “looks like”. When figures rotate, the relationships should survive. The student can turn the paper sideways and still find the same lengths.
How to find lengths that are not labelled
The missing lengths in the first example are not guesses. The full width is 12 cm, and the cut-out occupies 5 cm of that width. The remaining upper horizontal side is 12 − 5 = 7 cm. Similarly, the full height is 8 cm and the cut-out occupies 3 cm, leaving 8 − 3 = 5 cm of the right side.
I teach students to group all horizontal travels from the leftmost point to the rightmost point, and likewise all vertical travels from bottom to top. On a rectilinear boundary, the horizontal displacement left must eventually balance the displacement right, and the upward displacement must balance downward displacement. This is an excellent way to check missing segments.
| Known relationship | Unknown | Computation |
|---|---|---|
| Whole width 12 cm = left portion + cut-out width 5 cm | Top-left horizontal side | 12 − 5 = 7 cm |
| Whole height 8 cm = lower right side + removed height 3 cm | Lower right vertical side | 8 − 3 = 5 cm |
| Whole width 14 cm = 9 cm + unknown width | Remaining horizontal portion | 14 − 9 = 5 cm |
| Whole height 11 cm = 4 cm + unknown height | Remaining vertical portion | 11 − 4 = 7 cm |
For a more demanding diagram, I would mark a dotted line only as a temporary organiser. Dotted internal lines help compute missing lengths and area, but they are not automatically parts of the outside perimeter. The ability to choose when the line is mathematical scaffolding and when it is actual boundary is central.
Why an inward notch changes the perimeter
Now start with a new 12 cm by 8 cm rectangle. This time cut a 4 cm wide, 2 cm deep rectangular notch into the middle of the top edge, not at a corner. The notch removes an area of 4 × 2 = 8 cm², leaving 96 − 8 = 88 cm².
But the perimeter increases. The original top boundary loses a straight segment of length 4 cm. It gains a path down 2 cm, across 4 cm, and up 2 cm, for a new exposed length of 8 cm. The net increase is 8 − 4 = 4 cm, so the new perimeter is 44 cm.
| Operation | Area | Perimeter |
|---|---|---|
| Original rectangle 12 × 8 | 96 cm² | 40 cm |
| Remove centred top notch 4 × 2 | −8 cm² | The top 4 cm is replaced by 2 + 4 + 2 cm |
| Result | 88 cm² | 40 − 4 + 8 = 44 cm |
A student who repeats “cutting makes the perimeter smaller” has overgeneralised. I would put the corner-cut and centred-notch examples next to one another and ask for the one geometric feature that changes the result: the centred notch creates two inward side paths where the corner cut merely replaces portions of the existing boundary.
Can a shape have the same perimeter but less area?
Yes. The corner-cut example already demonstrated this: the original 12 × 8 rectangle has area 96 cm² and perimeter 40 cm; the corner-cut L-shape has area 81 cm² and the same 40 cm perimeter. Neither quantity determines the other by itself.
For a linked exploration, see Can Perimeter Increase While Area Decreases?. That broader conceptual question is related, but this lesson owns the missing-length and boundary-tracing method in PSLE-style composite figures.
A fully enclosed hole has its own boundary
Consider another 12 cm by 8 cm rectangle with a small 3 cm by 2 cm rectangular hole completely inside it. The material left has area 96 − 6 = 90 cm². The outer rectangular edge still measures 40 cm.
If the problem explicitly asks for the total length of all exposed boundaries, including the edge of the hole, we must add the inner perimeter: 2(3 + 2) = 10 cm. Total boundary length = 40 + 10 = 50 cm.
If instead the question specifically asks only for the outer perimeter, the answer is 40 cm. The wording matters. In Mathematics, “perimeter” convention and the diagram context must be read carefully for regions with holes; I would not assume the examiner means the total boundary without checking the exact instruction.
| Question | Boundary counted | Answer |
|---|---|---|
| Find outer perimeter | Outside rectangle only | 40 cm |
| Find perimeter of the rectangular hole | Hole’s four edges only | 10 cm |
| Find total length of all exposed edges | Outside plus hole boundary | 50 cm |
| Find remaining area of the region | Large area minus hole | 90 cm² |
I like this example because it prevents keyword-only Mathematics. The student has to read the quantity, define the region, and compute accordingly—the same skill later used in geometry, design, engineering and spatial modelling.
The tutor’s four-way geometry error diagnosis
A student may return from school with a wrong answer of 96, 81, 32 or 48. Looking only at the number does not tell me why. I ask for their first drawing and the precise line at which they stopped thinking about the region correctly.
| What I observe | Likely cause | Useful next test |
|---|---|---|
| Writes 96 cm² for the cut-corner L-shape | Forgot to subtract the removed material | Shade the region left and its cut-out separately |
| Uses cm² for boundary length | Conflates dimensions and units | Label a 1 cm line and a 1 cm² tile |
| Includes dotted construction lines in perimeter | Confuses interior organisers with edges | Trace only boundary with a finger |
| Omits a hidden horizontal or vertical side | Has not related parts to full dimensions | Recover lengths using total width or height |
| Claims every cut reduces perimeter | Transfers a memorised rule too widely | Contrast corner cut with centred notch |
When the child is correct after I draw the decomposition but not when they must choose it, the bottleneck is independent representation. When the model is accurate but the multiplication is wrong, the bottleneck is arithmetic execution. Those do not require identical tuition.
Try these PSLE-style composite-shapes problems
Question 1: a larger corner cut
A rectangle is 14 cm wide and 9 cm tall. A rectangle 6 cm wide and 4 cm tall is cut away from the top-right corner. Find the remaining area and perimeter.
Solution: Area = 14 × 9 − 6 × 4 = 126 − 24 = 102 cm². Because the cut is from a corner, its two removed boundary lengths are replaced by equal inward boundary lengths. Perimeter = 2(14 + 9) = 46 cm.
Question 2: a deeper notch
A 15 cm by 10 cm rectangle has a notch 5 cm wide and 3 cm deep removed from the middle of the top edge. Find the area and perimeter of the remaining region.
Solution: Area = 150 − 15 = 135 cm². Original perimeter = 50 cm. Replacing a five-centimetre top segment with down 3, across 5 and up 3 adds six centimetres. Perimeter = 56 cm.
Question 3: area and perimeter do not move together
Two shapes have perimeter 40 cm. Must they also have equal area? Give a counterexample using rectangles.
Solution: No. A 12 cm by 8 cm rectangle has perimeter 40 cm and area 96 cm². A 10 cm by 10 cm square also has perimeter 40 cm but area 100 cm². Their equal boundary totals do not force equal interiors.
Question 4: an inner boundary
A 10 cm by 9 cm rectangle contains a rectangular hole 2 cm by 3 cm completely inside. Find the remaining area and the total length of all exposed boundaries, including around the hole.
Solution: Remaining area = 90 − 6 = 84 cm². Outer perimeter = 2(10 + 9) = 38 cm. Inner boundary = 2(2 + 3) = 10 cm. Total exposed boundary length = 48 cm.
These four questions are deliberately related. If a child answers only the first, they may be copying the corner-cut method. If they solve the changed notch and inner-boundary questions, they are beginning to recognise the mathematical object rather than a particular diagram.
How to revise geometry before PSLE without overload
I would first revisit rectangle and square measurements, then teach one composite structure at a time. The next stage is mixed practice where the student chooses between adding pieces, subtracting missing pieces and reconstructing an overall boundary. Only after the method is secure would I make the task timed.
- Start with one clear diagram, not a cluttered multi-part past-year question.
- Ask the student to mark the measured region and physically trace the boundary.
- Draw missing horizontal and vertical totals with light construction lines.
- Work the area and perimeter separately even when the same picture is used.
- Change the shape, not only the dimensions, and ask whether the explanation still works.
- Return to the skill several days later without showing the answer key.
For the child’s exact Primary level, use the Ministry of Education’s current Primary Mathematics syllabus (updated October 2025) and the school’s teaching sequence. A conceptual model can be explored early, but it should not be confused with a claim about which specific problem types will appear in a future PSLE paper.
In tuition, ask how a teacher handles a student who solves area accurately but miscounts boundary. The useful approach is to retain the successful area skill and target the perimeter representation, instead of forcing the child to repeat every formula.
Frequently asked PSLE Maths geometry questions
Is an L-shaped figure always best solved by subtracting a rectangle?
Subtraction from an enclosing rectangle often works well for one corner cut. An L-shape can also be divided into non-overlapping rectangles and their areas added. The best route is the one the child can justify and check.
Why did the perimeter stay the same after the corner cut?
The two lengths removed from the outer rectangle are replaced by inward edges of the same total length. The result depends on the exact corner-cut geometry, not on a universal rule about removing material.
Does every indentation increase perimeter?
Not necessarily under every possible geometry and measurement definition. For a rectangular notch cut into the middle of a straight edge, the perimeter increases by twice the notch depth. Always trace the actual boundary.
Can a child use internal dotted lines to find missing lengths?
Yes. They can help extend dimensions and divide an area into pieces. The important distinction is that construction lines are not automatically perimeter edges.
How much PSLE Maths tuition is enough to fix composite figures?
That depends on whether the gap is units, spatial representation, missing-length logic, arithmetic or exam timing. Check for independent performance on changed shapes after a delay. No single worksheet count guarantees mastery or an examination score.
Continue through the Mathematics library
For a full Primary-to-JC route, open the Mathematics Learning Hub. To see the related conceptual issue, read Can Perimeter Increase While Area Decreases?. To strengthen early quantitative foundations, use P1–P4 Number Sense and Place Value. For different PSLE word-problem structures, visit PSLE Reference-Whole Problem Sums and PSLE Speed, Distance and Time.
To compare the right kind of support, read how to choose a Mathematics tutor in Singapore or arrange a parent–student consultation. Bring both the original diagram and the student’s first working so the teaching can start at the precise incorrect move.