Find compound area and follow only the exposed boundary
Split or subtract rectangles without double counting · Find missing boundary lengths · Exclude internal dividing lines from perimeter
Section Paper 3 · Historical Core · calculator
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
- Rectangle areas and perimeter
- Length subtraction
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
- Split or subtract rectangles without double counting
- Find missing boundary lengths
- Exclude internal dividing lines from perimeter
Two ways to the same area
For the L-shaped figure, begin with an 8 cm by 6 cm rectangle and remove the upper-right 3 cm by 2 cm rectangle. Area is 48−6 = 42 cm². Alternatively use a bottom 8 by 4 rectangle and a top-left 5 by 2 rectangle: 32+10 = 42 cm². Parts must neither overlap nor leave gaps.
Walk around the outside
Its boundary lengths are 8, 4, 3, 2, 5 and 6 cm, giving perimeter 28 cm. The dashed dividing line is inside the shape and is not counted. Do not add the perimeters of component rectangles unless their shared edges are removed twice.
This corner cut-out happens to leave the original rectangle perimeter unchanged. A cut-out in the middle of an edge or a hole can change the exposed boundary differently; inspect the particular geometry.
Sources for this lesson
Pause and explain
Should an internal dividing line be included in the outside perimeter?
Worked example
Find area and perimeter of the illustrated L shape.
Show the worked solution
- Area = 8×6−3×2 = 42 cm².
- Follow the six outer edges: 8+4+3+2+5+6 = 28 cm.
Answer 42 cm²; 28 cm.
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.
An 8 cm by 6 cm rectangle loses its upper-right 3 cm by 2 cm corner. Find the remaining area.
Give me a hint
Subtract the removed rectangle.
Compare my reasoning
- Outer area = 48 cm².
- Removed area = 6 cm²; remaining area = 42 cm².
42 cm².
Look for these in your work
- Use non-overlapping areas
- Keep square units
For that L shape, list all boundary lengths and find the perimeter.
Give me a hint
The shortened edges are 8−3 and 6−2.
Compare my reasoning
- Boundary lengths are 8, 4, 3, 2, 5, 6 cm.
- Their sum is 28 cm; omit the internal split.
28 cm.
Look for these in your work
- Recover missing lengths
- Trace the entire boundary
A 10 m by 6 m rectangular floor has a central 2 m by 2 m opening. Find floor area and the total exposed boundary length, including the opening.
Give me a hint
The hole removes area but adds an internal exposed boundary.
Compare my reasoning
- Floor area = 60−4 = 56 m².
- Outer boundary is 32 m; opening boundary is 8 m, making 40 m in total.
56 m²; total exposed boundary 40 m.
Look for these in your work
- Distinguish a hole from a corner cut
- Include the explicitly requested opening boundary
Common mistakes
- An internal split is not part of the outer perimeter.
- Do not assume every removed area changes perimeter in the same way.
What can you explain now?
Should an internal dividing line be included in the outside perimeter?
Compare with the explanation
No; it is not an exposed boundary.
A perimeter follows the outside edge of the region.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Should an internal dividing line be included in the outside perimeter?
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.