Unroll a cylinder and count its circular ends
Derive curved area from circumference times height · Distinguish closed, open-top and open-ended cylinders · Include the circular faces that are exposed
Section Paper 2 · Historical Extended · non-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
- Circle circumference and area
- Rectangle area
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
- Derive curved area from circumference times height
- Distinguish closed, open-top and open-ended cylinders
- Include the circular faces that are exposed
Unroll the curved sheet
The curved surface of a right circular cylinder unrolls to a rectangle of width 2πr and height h. Its area is 2πrh. A closed cylinder adds two circular ends: total area 2πrh+2πr². The curved rectangle width is a circumference, not the diameter.
Count ends explicitly
The radius-3 cm, height-8 cm diagram has curved area 48π cm² and two ends totalling 18π cm², giving closed area 66π cm². An open-top container adds only its bottom, making 57π cm². A thin tube open at both ends has just the curved area 48π cm² when edge thickness is ignored.
Sources for this lesson
Pause and explain
What is the width of a cylinder curved-surface net?
Worked example
Find the curved and total closed area for r=3 cm, h=8 cm.
Show the worked solution
- Curved area=2π×3×8=48π cm².
- Two ends=2π×3²=18π cm²; total=66π cm².
Answer Curved 48π cm²; closed total 66π 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.
Find curved and closed total area for r=2 cm and h=5 cm.
Give me a hint
Compute the curved sheet and two disks separately.
Compare my reasoning
- Curved=2π×2×5=20π cm².
- Two ends=8π cm²; total=28π cm².
Curved 20π cm²; closed total 28π cm².
Look for these in your work
- Count the required surfaces
- Use square units
An open-top thin cylinder has radius 4 cm and height 10 cm. Find its material area, ignoring thickness and overlaps.
Give me a hint
Add one bottom disk to the curved sheet.
Compare my reasoning
- Curved=80π cm².
- Bottom=16π cm²; total=96π cm².
96π cm².
Look for these in your work
- Count one end only
- Respect the stated model
For r=3 cm, h=8 cm, compare sheet areas of a tube open at both ends, an open-top bucket and a closed can. Ignore thickness.
Give me a hint
Each added end contributes 9π cm².
Compare my reasoning
- Curved sheet is 48π cm².
- Add zero, one or two ends, giving 48π, 57π and 66π cm².
Tube 48π cm²; bucket 57π cm²; closed can 66π cm².
Look for these in your work
- Identify actual end faces
- Separate area from capacity
Common mistakes
- Do not use diameter for the unrolled sheet width.
- An open top still has a bottom unless stated otherwise.
What can you explain now?
What is the width of a cylinder curved-surface net?
Compare with the explanation
2πr.
It wraps once around the circular circumference.
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Make it stick
What is the width of a cylinder curved-surface net?
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.