An L-shaped figure is drawn on a page.
Six side lengths are labelled.
Two are missing.
A learner immediately adds every visible number.
Another learner immediately splits the figure into rectangles and multiplies.
Both may be doing correct arithmetic.
Only one may be answering the question being asked.
For a rectilinear figure, perimeter follows the outside boundary; area covers the inside region. The figure can support both measurements, but the reasoning paths are different.
This article owns a narrower job than the general distinction between area and perimeter. The focus here is composite rectilinear figures: shapes made from horizontal and vertical line segments, often assembled from rectangles and squares, where missing side lengths and hidden decompositions must be inferred before calculation.
The updated October 2025 Singapore Primary Mathematics syllabus places, in Primary 4, finding the area and perimeter of composite figures made up of rectangles and squares, as well as finding an unknown dimension of a rectangle or square from area or perimeter. That makes rectilinear reasoning a natural bridge from simple formulas to structural geometry.
The quick answer: trace for perimeter, decompose for area
For perimeter:
- Trace the outer boundary only.
- Find any missing side lengths.
- Add each outside segment exactly once.
- Use a linear unit such as cm or m.
For area:
- Identify rectangles or squares that cover the region without overlap.
- Find any missing dimensions.
- Calculate each rectangular area.
- Add the parts, or subtract a missing rectangle from a larger enclosing rectangle.
- Use square units such as cm² or m².
These two routines solve different measurement jobs.
What makes a figure rectilinear?
A rectilinear figure is bounded by straight horizontal and vertical segments meeting at right angles.
An L-shape is a common example.
A staircase-shaped polygon is another.
Because every turn is a right angle, horizontal distances and vertical distances can often be balanced across the figure.
This creates useful hidden equalities.
For example, the total horizontal distance travelled to the right around a closed rectilinear boundary must equal the total horizontal distance travelled to the left.
Likewise, total upward distance equals total downward distance.
That fact is one of the cleanest ways to find missing perimeter lengths.
Perimeter is a closed journey
Imagine walking around the edge of the figure.
You start at one corner and return to the same corner.
The perimeter is the total distance walked.
This travel interpretation is powerful because it prevents two common errors:
- including internal partition lines that are not part of the outside edge;
- forgetting short “inward” or “outward” segments in an L-shaped boundary.
A finger trace or pencil trace around the outside can be more reliable than looking at the figure all at once.
Finding a missing horizontal length
Suppose an L-shaped figure has a total width of 12 cm across the top.
Along the lower broken path, two horizontal sections measure 5 cm and an unknown length x.
If those two sections together span the same total left-to-right width, then:
5 + x = 12.
So:
x = 7 cm.
The missing side was not guessed from visual proportions.
It was derived from horizontal balance.
Finding a missing vertical length
Suppose total height is 9 cm.
One vertical section is 4 cm and the remaining vertical section is y.
Then:
4 + y = 9.
So:
y = 5 cm.
Again, the shape supplies an equality before it supplies a formula.
A stronger perimeter shortcut: opposite-direction totals balance
For any closed rectilinear path:
- total rightward length = total leftward length;
- total upward length = total downward length.
Why?
Because returning to the starting point requires zero net horizontal displacement and zero net vertical displacement.
Primary 4 learners do not need vector notation to use the idea.
They can reason:
“Whatever distance I travelled right must eventually be balanced by the same total distance left.”
This can reveal missing sides quickly and provides a strong checking method after calculation.
Area method 1: split the figure into rectangles
Take an L-shaped floor plan.
One common method is to draw a dividing line that turns the L into two rectangles.
Suppose Rectangle A is 8 cm by 4 cm.
Area A = 32 cm².
Rectangle B is 3 cm by 5 cm.
Area B = 15 cm².
Total area:
32 + 15 = 47 cm².
The dividing line is not part of the perimeter.
It is a temporary reasoning tool used only to decompose the area.
Area method 2: complete a larger rectangle and subtract
The same L-shape can often be viewed as:
large rectangle − missing corner rectangle.
Suppose the enclosing rectangle measures 11 cm by 9 cm.
Area = 99 cm².
The missing rectangular notch measures 4 cm by 3 cm.
Missing area = 12 cm².
Target area:
99 − 12 = 87 cm².
This method can be faster when the outer dimensions are already given.
Different decompositions should give the same area
A useful diagnostic task is to solve the same rectilinear area in two ways.
- Method A: split vertically into rectangles.
- Method B: split horizontally into rectangles.
- Method C: large enclosing rectangle minus missing region.
If all decompositions describe exactly the same region without gaps or overlaps, all should produce the same total area.
This is a powerful idea:
Area is invariant under valid decomposition. The lines we draw to organise the calculation do not change the region being measured.
Worked example: perimeter of an L-shaped figure
Suppose a rectilinear figure has these boundary segments in clockwise order:
12 cm right, 4 cm down, 5 cm left, 5 cm down, 7 cm left, 9 cm up.
Check closure:
Rightward total = 12 cm.
Leftward total = 5 + 7 = 12 cm.
Downward total = 4 + 5 = 9 cm.
Upward total = 9 cm.
The path closes.
Perimeter:
12 + 4 + 5 + 5 + 7 + 9 = 42 cm.
Worked example: missing side before perimeter
Total top width = 15 cm.
Two lower horizontal segments are 6 cm and x cm.
Then x = 15 − 6 = 9 cm.
Only after finding x should it be inserted into the perimeter sum.
A common error is to add the visible side labels first and discover too late that the boundary is incomplete.
Worked example: area from an enclosing rectangle
A rectilinear figure fits inside a 14 m by 10 m rectangle.
A 5 m by 4 m rectangular corner is missing.
Large rectangle:
14 × 10 = 140 m².
Missing rectangle:
5 × 4 = 20 m².
Target area:
140 − 20 = 120 m².
Unknown dimensions from area
Suppose a rectangle has area 72 cm² and one side 8 cm.
The missing side is:
72 ÷ 8 = 9 cm.
This reverse use of the area formula is important in composite figures because a missing sub-rectangle dimension may need to be recovered before decomposition can continue.
Unknown dimensions from perimeter
A rectangle has perimeter 34 cm and length 10 cm.
Two lengths contribute:
10 + 10 = 20 cm.
The two widths together contribute:
34 − 20 = 14 cm.
One width:
14 ÷ 2 = 7 cm.
Again, the formula can be used backwards when the unknown changes position.
Area and perimeter can react differently to the same notch
Cutting a rectangular notch from a larger rectangle reduces area.
Its effect on perimeter depends on where the notch is cut.
If the notch is removed from a corner, some old boundary segments disappear and new shorter boundaries replace them.
If a notch is cut into the middle of one side, new inward edges can increase the perimeter even though area decreases.
This is a useful transfer reminder: area and perimeter do not move in lockstep.
Common misconception 1: every line in the diagram belongs to the perimeter
Internal decomposition lines are not part of the outside boundary.
Repair: trace the perimeter physically before adding any lengths.
Common misconception 2: all labelled lengths should be multiplied for area
Area requires dimensions belonging to the same rectangle or square.
Multiplying unrelated side labels creates meaningless products.
Repair: outline each chosen rectangle first, then identify its own length and width.
Common misconception 3: the drawing is to scale
School diagrams are often schematic.
A side drawn longer may represent a shorter labelled length.
Repair: trust stated dimensions and geometric relationships, not appearance.
Common misconception 4: square units are optional
Area measures two-dimensional coverage.
Use cm², m² and other square units.
Perimeter remains in linear units such as cm or m.
Common misconception 5: there is one correct decomposition
Many rectilinear figures can be decomposed in several valid ways.
Repair: compare two decompositions and verify that both cover the target region exactly once.
A diagnostic ladder for rectilinear figures
- Can the learner distinguish perimeter from area?
- Can the learner trace only the outer boundary?
- Can the learner identify internal lines that should not enter the perimeter?
- Can the learner find a missing horizontal length from total width?
- Can the learner find a missing vertical length from total height?
- Can the learner use right/left and up/down balance as a check?
- Can the learner split an L-shape into rectangles?
- Can the learner use an enclosing rectangle minus a missing part?
- Can the learner recover an unknown dimension from area or perimeter?
- Can the learner maintain correct linear or square units?
- Can the learner solve the same area using two valid decompositions and obtain the same result?
A five-minute home investigation
Draw a 6-by-4 rectangle on squared paper.
Remove a 2-by-2 corner to create an L-shape.
Ask:
- What happened to the area?
- What happened to the perimeter?
- Can the area be found by splitting into rectangles?
- Can it also be found by 6×4 − 2×2?
- Which new boundary segments appeared after the corner was removed?
Then move the 2-by-2 cutout from the corner to the middle of one side and ask how the perimeter changes.
This turns a worksheet topic into a small geometric experiment.
What parents should listen for
- “I am tracing only the outside because the question asks for perimeter.”
- “These two lower horizontal parts must add to the full top width.”
- “I split the area into two rectangles, and the dividing line does not change the region.”
- “I used cm² for area but cm for perimeter.”
- “My second decomposition gives the same area, so the first method is probably sound.”
What teachers and tutors should avoid
- Avoid formula-first teaching before boundary and region are distinguished.
- Avoid giving every missing side explicitly. Let horizontal and vertical totals carry some of the reasoning.
- Avoid treating diagrams as necessarily drawn to scale.
- Avoid requiring one decomposition when several are valid.
- Avoid marking a correct numeral as fully correct when the unit is wrong.
- Avoid using the broad area/perimeter distinction page as a substitute for composite-figure reasoning. This narrower job requires missing-length and decomposition work.
How this fits Singapore Primary 4 Mathematics
The updated October 2025 MOE Primary Mathematics syllabus includes, in Primary 4, finding one dimension of a rectangle when the other dimension and area or perimeter are known, finding the side of a square from area or perimeter, and finding area and perimeter of composite figures made up of rectangles and squares.
The broader concept that area and perimeter measure different things remains foundational. The Primary 4 upgrade is structural: the figure may no longer be one simple rectangle, and the learner must create a useful representation before calculating.
The deeper lesson: decomposition creates solvable pieces without changing the whole
A complicated rectilinear figure can feel unfamiliar.
Yet it is often built from familiar rectangles.
The mathematical move is to change the representation while preserving the measured object.
For area, we partition the region into manageable pieces.
For perimeter, we linearise the boundary into a sequence of segments.
Composite geometry becomes easier when the learner asks what can be decomposed and what must remain invariant.
Final thought
An L-shape is not difficult because it needs a special L-shape formula.
It is difficult because the useful rectangle or boundary path is not given to the learner in finished form.
Once the learner can expose that structure, ordinary length, addition, subtraction and multiplication do the rest.
Trace what surrounds the figure. Decompose what fills it.