Calculate rectangle and triangle areas with perpendicular heights
Distinguish boundary length from enclosed area · Use base times perpendicular height · Avoid using a sloping side as a height
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Length units
- Right angles
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Distinguish boundary length from enclosed area
- Use base times perpendicular height
- Avoid using a sloping side as a height
Boundary and interior
A rectangle with sides l and w has perimeter 2(l+w) and area lw. A triangle has perimeter equal to the sum of its three sides, but area (1/2)bh, where h is perpendicular to the chosen base b. Area uses square units; perimeter uses length units.
The perpendicular height may fall outside an obtuse triangle. The area rule still works; it does not require the height segment to be inside the triangle.
Read the actual height
For a triangle with base 8 cm and perpendicular height 5 cm, area is 20 cm². A side of length 6 cm cannot replace that height unless it is perpendicular to the selected base. A diagram that looks nearly perpendicular is not sufficient evidence; use stated right-angle information.
Sources for this lesson
Pause and explain
A triangle has base 8 cm and perpendicular height 5 cm. What is its area?
Worked example
A rectangle is 8 cm by 5 cm. Find perimeter and area.
Show the worked solution
- Perimeter = 2(8+5) = 26 cm.
- Area = 8×5 = 40 cm².
Answer 26 cm; 40 cm².
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A rectangle is 7 m by 3 m. Find its perimeter and area.
Give me a hint
Calculate the two quantities separately.
Compare my reasoning
- Perimeter = 2(7+3) = 20 m.
- Area = 7×3 = 21 m².
20 m; 21 m².
Look for these in your work
- Name the quantity
- Use the right units
A triangle has base 10 cm, perpendicular height 4 cm and a sloping side 6 cm. Find its area.
Give me a hint
Select the perpendicular height, not the side.
Compare my reasoning
- Area = (1/2)×10×4.
- The sloping 6 cm side is not needed.
20 cm².
Look for these in your work
- Identify the base-height pair
- Reject the wrong side
Two rectangular plots measure 12 m by 3 m and 9 m by 4 m. Do they have equal area and require equal boundary fencing?
Give me a hint
Compare area and perimeter separately.
Compare my reasoning
- Both areas are 36 m².
- Perimeters are 30 m and 26 m, so the fencing differs.
Equal area 36 m²; different perimeters 30 m and 26 m.
Look for these in your work
- Check both quantities
- Do not infer perimeter from area
Common mistakes
- A side is not automatically a perpendicular height.
- Area and perimeter have different units.
What can you explain now?
A triangle has base 8 cm and perpendicular height 5 cm. What is its area?
Compare with the explanation
20 cm².
The corresponding base-by-height rectangle has area 8×5=40 cm²; the triangle has half that area.
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Make it stick
A triangle has base 8 cm and perpendicular height 5 cm. What is its area?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.