PSLE Mathematics Learning Guide · Guide 50
Return to the PSLE Learning Guide · Mathematics Learning Hub
Parallelogram and trapezium area problems become much easier when the learner separates side length from perpendicular height.
A sloping side may be longer than the height. A base may be drawn at the top instead of the bottom. A trapezium may be rotated. None of these changes the fundamental area relationships.
This guide builds one habit: identify the parallel sides, choose a valid base, find the perpendicular height between the parallel lines, and decompose or rearrange the figure when the formula is not immediately obvious.
Guide 26 owns triangle area and perpendicular height. Guide 6 owns composite area/perimeter. This page focuses on parallelograms and trapeziums as distinct area structures while routing back to those owners where needed.
The MOE Primary Mathematics syllabus updated October 2025 provides current curriculum context. All examples below are original eduKate teaching material.
A parallelogram can be rearranged into a rectangle
Imagine a parallelogram with base 8 cm and perpendicular height 5 cm. Cut a triangular piece from one side and move it to the other side. The shape becomes a rectangle measuring 8 cm by 5 cm.
No area has been added or removed.
Therefore:
Area of parallelogram = base × perpendicular height.
For the example: 8 × 5 = 40 cm².
The sloping side is not automatically the height
A parallelogram has base 10 cm, sloping side 7 cm and perpendicular height 6 cm.
Area = 10 × 6 = 60 cm².
Using 10 × 7 would multiply the base by a non-perpendicular side and overstate the area.
Height measures shortest perpendicular distance between the parallel base lines.
Either pair of opposite parallel sides can act as bases
A parallelogram can be described using one side as the base and the matching perpendicular distance as its height.
If a different side is chosen as base, the matching height changes. The area remains the same.
This is the same geometric idea used in triangle area: base and height must be a valid perpendicular pair.
Find a missing height from area
Area = 72 cm², base = 9 cm.
Height = 72 ÷ 9 = 8 cm.
Check: 9 × 8 = 72 cm².
Find a missing base from area
Area = 84 cm², height = 7 cm.
Base = 84 ÷ 7 = 12 cm.
A trapezium has one pair of parallel sides in the school convention used here
The area depends on the lengths of the two parallel sides and the perpendicular distance between them.
Let the parallel sides be 8 cm and 14 cm, with perpendicular height 5 cm.
Area = 1/2 × (8 + 14) × 5.
= 1/2 × 22 × 5.
= 55 cm².
Why the trapezium formula works
Take two congruent copies of the same trapezium and rotate one. They can be arranged into a parallelogram whose base is the sum of the two parallel sides and whose height is the same perpendicular height.
The parallelogram area is:
(a + b) × h.
One trapezium is half of that:
Area = 1/2 × (a + b) × h.
This meaning is more reliable than memorising a string of symbols with no geometric picture.
The trapezium formula can also be read as average parallel side × height
For parallel sides 8 cm and 14 cm:
average parallel-side length = (8 + 14) ÷ 2 = 11 cm.
Area = 11 × 5 = 55 cm².
This is mathematically equivalent to the half-sum formula.
Height is perpendicular distance between the parallel sides
If a trapezium has sloping sides 6 cm and 7 cm, neither is automatically the height.
The height must be perpendicular to both parallel sides because perpendicular distance between two parallel lines is constant.
A right-angle mark or a perpendicular construction can identify it.
A trapezium can be split into simpler shapes
Suppose a trapezium has parallel sides 12 cm and 20 cm, height 6 cm.
One decomposition is a 12×6 rectangle plus a right triangle with base 8 cm and height 6 cm.
Rectangle area = 72 cm².
Triangle area = 1/2×8×6 = 24 cm².
Total = 96 cm².
Formula check:
1/2×(12+20)×6 = 16×6 = 96 cm².
Another trapezium can be split into two triangles
Draw a diagonal from one vertex to the opposite vertex. The trapezium becomes two triangles that share the same perpendicular height to the parallel sides.
Their areas add to:
1/2×a×h + 1/2×b×h = 1/2×(a+b)×h.
This shows the formula through addition rather than rearrangement.
Find trapezium height from area
Area = 90 cm², parallel sides 10 cm and 20 cm.
90 = 1/2×30×h = 15h.
h = 6 cm.
Find one missing parallel side
Area = 84 cm², height 7 cm, one parallel side 8 cm. Let the other parallel side be b.
84 = 1/2×(8+b)×7.
168 = 7(8+b).
24 = 8+b.
b = 16 cm.
Check: average parallel side = 12 cm; 12×7 = 84 cm².
Rotation does not change the formula
A trapezium drawn sideways still has two parallel sides and a perpendicular distance between them. The words “top” and “bottom” are convenient but not mathematical requirements.
Likewise, a parallelogram drawn on a slant does not need a horizontal base.
Use properties, not page orientation.
Parallelograms with the same base and height have equal area
If several parallelograms share base 12 cm and perpendicular height 5 cm, each has area 60 cm², even if their top sides are shifted sideways.
The slant changes shape but not base or perpendicular height.
Triangle and parallelogram areas are connected
A triangle and a parallelogram share the same base 10 cm and height 6 cm.
Parallelogram area = 60 cm².
Triangle area = 1/2×10×6 = 30 cm².
The triangle has half the area of the parallelogram when base and height match.
Composite figures may hide a trapezium or parallelogram inside
A complex polygon can sometimes be split into a rectangle and trapezium, or into a triangle and parallelogram.
The key is to choose a decomposition that uses known dimensions and avoids creating unnecessary unknowns.
Use Guide 6 when decomposition of the whole figure is the main job.
Main worked workshop: sixteen original area tasks
1.
Parallelogram base 9 cm, height 4 cm.
Answer: 36 cm².
2.
Parallelogram base 15 cm, height 7 cm.
Answer: 105 cm².
3.
Parallelogram area 96 cm², base 12 cm. Height?
Answer: 8 cm.
4.
Parallelogram area 91 cm², height 7 cm. Base?
Answer: 13 cm.
5.
Base 12 cm, sloping side 10 cm, perpendicular height 8 cm. Area?
Answer: 96 cm².
6.
Trapezium parallel sides 6 cm and 14 cm, height 5 cm.
Answer: 50 cm².
7.
Trapezium parallel sides 9 cm and 15 cm, height 8 cm.
Answer: 96 cm².
8.
Trapezium area 72 cm², parallel sides 8 and 16 cm. Height?
Answer: 6 cm.
9.
Trapezium area 105 cm², height 7 cm, one parallel side 10 cm. Other side?
Reasoning: average side = 105÷7=15; sum =30; other side=20 cm.
10.
A learner uses the 7 cm sloping side as height in a parallelogram whose perpendicular height is 5 cm.
Repair: use perpendicular height 5 cm.
11.
A learner uses all four trapezium side lengths in the area formula.
Repair: area requires only the two parallel sides and perpendicular height.
12.
Two parallelograms both have base 11 cm and height 6 cm but different slants.
Answer: both have area 66 cm².
13.
A trapezium has parallel sides 10 and 18 cm, height 4 cm.
Answer: average side 14; area 56 cm².
14.
A 20×8 rectangle has a right triangle of base 6 and height 8 removed, leaving a trapezium.
Answer: rectangle 160 − triangle 24 = 136 cm².
15.
A triangle and parallelogram share base 14 cm and height 9 cm. Compare their areas.
Answer: parallelogram 126 cm²; triangle 63 cm²; ratio 2:1.
16.
Trapezium parallel sides differ by 8 cm, smaller side 12 cm, height 5 cm.
Answer: larger side 20; area = 1/2×32×5 = 80 cm².
Area needs square units
Base and height may be measured in centimetres, metres or another length unit. After multiplication, area uses square units such as cm² or m².
If dimensions use different units, convert them before applying the area relationship.
Use bounding shapes to check size
A trapezium with parallel sides 8 cm and 14 cm and height 5 cm has area 55 cm².
It should be larger than an 8×5 rectangle area of 40 cm² and smaller than a 14×5 rectangle area of 70 cm².
55 lies between those bounds, so the answer is plausible.
This check follows from the trapezium’s average parallel-side length lying between the two side lengths.
Common area errors
Error 1: use a sloping side instead of perpendicular height.
Error 2: forget the 1/2 in trapezium area.
Error 3: average the wrong pair of sides.
Error 4: use perimeter information as though it directly gives area.
Error 5: write a length unit instead of a square unit.
Independent transfer check
- Parallelogram base 18 cm, height 5 cm. Find area.
- Parallelogram area 132 cm², base 11 cm. Find height.
- Trapezium parallel sides 7 and 13 cm, height 6 cm. Find area.
- Trapezium area 96 cm², height 8 cm, one parallel side 10 cm. Find the other.
- Explain why a sloping side is not automatically a height.
- Explain why two parallelograms with equal base and height can have equal area despite different slants.
- Use the average-parallel-side form to find area when parallel sides are 12 and 20 cm, height 7 cm.
- Give one reason a trapezium area answer should lie between the areas of rectangles using the shorter and longer parallel sides with the same height.
Independent-check answers
1. 90 cm².
2. 12 cm.
3. 60 cm².
4. 14 cm.
5. Height is defined by perpendicular distance to the chosen base or parallel sides.
6. Area depends on base and perpendicular height; sideways shift changes slant, not those dimensions.
7. Average side 16 cm; area 112 cm².
8. The trapezium’s average parallel-side length lies between the two side lengths.
Parent and tutor guide
Before allowing a formula, ask the learner to point to the two parallel sides and draw the perpendicular height. Rotate diagrams often so “bottom” does not become a hidden rule.
Use a paper parallelogram cut-and-shift activity to show why base×height works. For trapeziums, pair two congruent copies into a parallelogram or decompose one into a rectangle and triangle.
After a direct area task, reverse the relationship. Ask for missing height or missing parallel side so the learner sees the formula as a connected system rather than a one-way instruction.
The learner’s final card
Which sides are parallel? What is the perpendicular height? Is this a parallelogram or trapezium? Can I rearrange or decompose it? Do my units become square units? Does the answer lie in a sensible size range?
Continue through Batch 13
Use Guide 49. Continue with Guide 51: Remainders in Context and Guide 52: Changing Averages.
Return to the PSLE Learning Guide Mathematics route.
Sources and boundaries
MOE Primary Mathematics syllabus, updated October 2025; SEAB 2026 PSLE formats.
All figures, dimensions and worked solutions are original eduKate teaching material. This guide focuses on area structure and routes to the established triangle and composite-area owners for adjacent reasoning.