PSLE Mathematics Learning Guide · Guide 26
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The formula for triangle area is familiar:
Area = 1/2 × base × height.
The difficulty is deciding which two lengths are allowed to play the roles of base and height. The height must be perpendicular to the chosen base. It is not automatically a sloping side, the longest side, or the line that looks most vertical on the page.
This guide uses one habit: choose a base, locate or reconstruct the perpendicular height to that base, apply the area relationship, then reverse the formula carefully when the base or height is unknown.
The MOE Primary Mathematics syllabus updated October 2025 provides curriculum context. The SEAB 2026 PSLE examination-format page links the current Mathematics syllabus. All examples below are original eduKate teaching material.
Why the factor one half appears
Two congruent copies of a triangle can often be arranged to form a parallelogram with the same base and perpendicular height. The parallelogram area is base × height, so one triangle has half that area.
This meaning is stronger than memorising the symbol 1/2 because it explains why the factor is present.
Any side can be a base if the matching perpendicular height is used
A triangle does not have one permanent base. If a different side is chosen as the base, the corresponding height changes.
The area must remain the same because the triangle has not changed. This gives an important check: different valid base-height pairs should produce equal area.
Height means perpendicular distance
If a triangle has a horizontal base, a vertical segment from the opposite vertex to the base line may represent the height. But the page orientation is not the definition. Perpendicularity is.
A right-angle mark, rectangle structure, grid alignment or known perpendicular relation can establish the height.
The perpendicular height can fall outside the triangle
For an obtuse triangle, the perpendicular from a vertex to the line containing the opposite side may meet an extension of that side outside the visible triangle.
The area formula still uses the perpendicular distance to the base line.
Do not reject a valid height merely because it lies outside the triangular region.
Right triangles provide an immediate base-height pair
The two sides forming the right angle are perpendicular, so either can be the base while the other is the height.
Example: legs 6 cm and 8 cm.
Area = 1/2 × 6 × 8 = 24 cm².
Find height from area and base
A triangle has area 54 cm² and base 12 cm.
54 = 1/2 × 12 × h = 6h.
h = 9 cm.
A useful direct form is:
height = 2 × area ÷ base.
Find base from area and height
A triangle has area 42 cm² and height 7 cm.
42 = 1/2 × b × 7.
84 = 7b.
b = 12 cm.
Direct form:
base = 2 × area ÷ height.
Triangles with the same base and same height have equal area
If two triangles share a 10 cm base and both have perpendicular height 6 cm, each has area:
1/2 × 10 × 6 = 30 cm².
The top vertex can move sideways along a line parallel to the base without changing the height, so the area remains 30 cm².
This invariant is powerful in diagram problems.
Same height means area follows the base length
Two triangles have the same perpendicular height. Their bases are 4 cm and 10 cm. Their areas are in ratio 4:10 = 2:5.
You do not need to know the height numerically to compare their areas.
Triangle area often appears inside composite figures
A rectangle 12 cm by 8 cm has a triangular corner removed with base 4 cm and perpendicular height 3 cm.
Rectangle area = 96 cm².
Triangle area = 1/2 × 4 × 3 = 6 cm².
Remaining area = 90 cm².
Guide 6 covers the wider decomposition problem: Build Composite Area and Perimeter From the Pieces That Actually Count.
Scaling base or height changes area predictably
If the base doubles and the height stays fixed, area doubles.
If the height triples and the base stays fixed, area triples.
If both base and height double, area becomes four times as large.
These checks help detect missing factors or wrong dimensions.
Main worked workshop: sixteen original triangle-area problems
1.
Base 10 cm, height 6 cm.
Answer: 30 cm².
2.
Base 15 cm, height 8 cm.
Answer: 60 cm².
3.
Right triangle legs 9 cm and 12 cm.
Answer: 54 cm².
4.
Area 36 cm², base 9 cm. Find height.
Answer: 8 cm.
5.
Area 70 cm², height 10 cm. Find base.
Answer: 14 cm.
6.
Area 45 cm², base 18 cm. Find height.
Answer: 5 cm.
7.
Base doubles from 7 to 14 cm while height stays fixed. What happens to area?
Answer: doubles.
8.
Both base and height double.
Answer: area becomes 4 times as large.
9.
Two triangles share the same 12 cm base and have equal perpendicular height. Compare areas.
Answer: equal.
10.
Two triangles have same height and bases 3 cm and 11 cm. Area ratio?
Answer: 3:11.
11.
A learner uses a 13 cm sloping side as height to a 10 cm base, although no perpendicular relation is given.
Repair: A height must be perpendicular to the chosen base.
12.
A triangle has base 16 cm and perpendicular height 7.5 cm.
Answer: 60 cm².
13.
A 20×12 rectangle has a triangular piece of base 8 and height 5 removed.
Answer: 240 − 20 = 220 cm².
14.
Triangle area is 96 cm², base 24 cm.
Answer: height 8 cm.
15.
Triangle area is 63 cm², height 9 cm.
Answer: base 14 cm.
16.
A learner calculates 1/2 × base × a non-perpendicular side and gets 40 cm².
Repair: The arithmetic may be neat but the input length is not a valid height. Re-identify the perpendicular distance before calculating.
Run a geometry audit before using the formula
- Which side am I calling the base?
- Where is the opposite vertex?
- What segment or distance is perpendicular to the base line?
- Is the height labelled, derivable, or still unknown?
- Does the answer require square units?
Common triangle-area errors
Error 1: Forget the factor 1/2.
Error 2: Use any side as height without perpendicularity.
Error 3: Multiply all three side lengths.
Error 4: Use perimeter information as though it directly gives height.
Error 5: Give cm instead of cm² for area.
Independent transfer check
- Base 14 cm, height 9 cm.
- Area 84 cm², base 12 cm. Find height.
- Area 50 cm², height 5 cm. Find base.
- Two triangles have equal height and bases 6 cm and 15 cm. Compare their areas.
- A right triangle has perpendicular sides 8 cm and 15 cm. Find area.
- Explain why a sloping side is not automatically a height.
- Explain why two triangles with the same base and height can have differently placed top vertices but equal area.
- If both base and height are tripled, by what factor does area change?
Independent-check answers
1. 63 cm².
2. 14 cm.
3. 20 cm.
4. 6:15 = 2:5.
5. 60 cm².
6. Height is defined by perpendicular distance to the chosen base.
7. Area depends only on base length and perpendicular height for this comparison.
8. 9 times.
Parent and tutor guide
Before giving the formula, ask the learner to point to the chosen base and draw or identify the perpendicular height. If the diagram rotates, keep the same definition rather than relying on “bottom” and “vertical”.
For reverse problems, write the area equation first and solve for the missing measure. Then substitute it back into the original formula.
The learner’s final card
Which side is my base? What is perpendicular to it? Am I finding area, base or height? Does my unit match the quantity? Can I reconstruct the area to check?
Continue through the PSLE Mathematics Learning Guide
Return to Guide 25: Elapsed Time and Timetables. Continue with Guide 27: Compare and Order Fractions and Guide 28: Money and Change.
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. Triangle-area reasoning is treated as cumulative Primary Mathematics geometry relevant to PSLE preparation.