Find polygon perimeters and triangle areas from coordinates
Add all polygon side lengths · Establish the right angle before using two sides as base and height · Recognise when collinear points give zero triangle area
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Distance formula
- Right-triangle area and converse Pythagoras
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
- Add all polygon side lengths
- Establish the right angle before using two sides as base and height
- Recognise when collinear points give zero triangle area
Find each side, then add
A perimeter is the sum of the boundary segment lengths, not a diagonal or a sum of coordinate values. For A(0,0), B(4,2), C(3,4), D(−1,2), adjacent sides have lengths √20 and √5. Their opposite displacements reverse, so the shape is a parallelogram.
Check perpendicular height
AC²=25 and AB²+BC²=20+5=25, so the converse of Pythagoras establishes a right angle at B. Thus this parallelogram is a rectangle: area √20×√5=10 square units, and perimeter 2(√20+√5)=6√5 units.
For a triangle, (1/2)ab is an area only when a and b are perpendicular, or when b is the perpendicular height to base a. Collinear points do not make a non-degenerate triangle; their enclosed area is zero.
Sources for this lesson
Pause and explain
Can any two triangle side lengths be multiplied and halved to obtain area?
Worked example
Find the area and perimeter of the illustrated rectangle ABCD.
Show the worked solution
- Adjacent side lengths are 2√5 and √5; the diagonal-square check establishes perpendicularity.
- Area=10; perimeter=2(2√5+√5)=6√5.
Answer Area 10 square units; perimeter 6√5 units.
Try a topic test
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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.
Find perimeter and area of triangle A(0,0), B(4,0), C(0,3).
Give me a hint
The axis-aligned legs are perpendicular.
Compare my reasoning
- AB=4, AC=3 and BC=5, so perimeter=12.
- Area=(1/2)×4×3=6.
Perimeter 12 units; area 6 square units.
Look for these in your work
- Use all three side lengths
- Use perpendicular legs for area
For the illustrated A(0,0), B(4,2), C(3,4), D(−1,2), justify that it is a rectangle and find its measures.
Give me a hint
Use side lengths and the diagonal-square check.
Compare my reasoning
- Adjacent side squares are 20 and 5; AC²=25 establishes the right angle. Opposite sides are parallel by reversed displacements.
- Area=√20√5=10; perimeter=2(√20+√5)=6√5.
Rectangle; area 10 square units; perimeter 6√5 units.
Look for these in your work
- Establish the geometric condition
- Keep exact lengths
Can A(0,0), B(2,1), C(4,2) enclose a positive-area triangle? Explain using their coordinate direction.
Give me a hint
Compare the two successive displacements.
Compare my reasoning
- A→B and B→C are both (2,1), so the three points are collinear.
- There is no enclosed triangular region; area is zero.
No; the collinear points give area 0.
Look for these in your work
- Identify degenerate geometry
- Do not invent a perpendicular height
Common mistakes
- Do not assume two sloping sides are a base-height pair.
- Include every boundary segment in perimeter.
What can you explain now?
Can any two triangle side lengths be multiplied and halved to obtain area?
Compare with the explanation
No; they must be perpendicular, or one must be the perpendicular height.
A sloping side generally differs from the height to the selected base.
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Make it stick
Can any two triangle side lengths be multiplied and halved to obtain area?
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