Add compound volumes and omit internal joining faces
Add volumes of non-overlapping joined solids · Identify surfaces hidden at a join · Distinguish a closed hemisphere from its exposed curved half
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
- Cylinder and sphere formulae
- Surface face accounting
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 volumes of non-overlapping joined solids
- Identify surfaces hidden at a join
- Distinguish a closed hemisphere from its exposed curved half
Separate volume from exposed area
A cylinder joined to a hemisphere of the same radius has volume πr²h+(2/3)πr³: the hemisphere is half the sphere volume. The pieces meet at one circular interface without overlapping volumes. If their interiors overlap, simply adding complete volumes would double count that overlap.
The interface is internal
For an exterior coating, include the cylinder curved surface 2πrh, the hemisphere curved surface 2πr² and the exposed bottom disk πr². Omit both copies of the joining disk. The exterior total is 2πrh+3πr². A standalone closed hemisphere would include its own flat circular base; joining it hides that base.
The radius-3 cm cylinder of height 5 cm with hemispherical cap has total volume 63π cm³ and exposed area 57π cm². The illustration is an axial section; use the three-dimensional formulae rather than the area of its outline.
Sources for this lesson
Pause and explain
How many copies of the joining disk remain on the exterior of the joined solid?
Worked example
Find volume and exposed area of the illustrated cylinder with hemispherical cap, r=3 cm and cylinder h=5 cm.
Show the worked solution
- Volume=45π+18π=63π cm³.
- Exterior area=30π+18π+9π=57π cm².
Answer 63π cm³; 57π cm².
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.
A cylinder r=3 cm, h=5 cm has a hemispherical cap of radius 3 cm. Find the total volume.
Give me a hint
Add the full cylinder to half a sphere.
Compare my reasoning
- Cylinder=45π cm³; cap=(2/3)π×27=18π cm³.
- Total=63π cm³.
63π cm³.
Look for these in your work
- Use half the sphere volume
- Avoid overlapping regions
Find that solid's exposed area, including the cylinder bottom.
Give me a hint
Keep the curved parts and one exposed disk.
Compare my reasoning
- Cylinder curve=30π cm²; cap curve=18π cm².
- Bottom=9π cm²; total=57π cm².
57π cm².
Look for these in your work
- Omit the hidden interface
- Include the exposed bottom
Adding the areas of a closed cylinder and a closed hemisphere gives 75π cm² for this model. Explain how to correct it.
Give me a hint
Each closed piece includes a joining disk of area 9π.
Compare my reasoning
- Remove both internal disk copies: 2×9π=18π cm².
- 75π−18π=57π cm².
Subtract 18π cm² to obtain 57π cm².
Look for these in your work
- Explain double counting
- Check the exposed-face sum
Common mistakes
- Two complete surface-area formulae include the joining disk twice.
- Do not use a section outline as a 3D surface area.
What can you explain now?
How many copies of the joining disk remain on the exterior of the joined solid?
Compare with the explanation
None.
Both contacting disk faces are internal.
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Make it stick
How many copies of the joining disk remain on the exterior of the joined solid?
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.