Subtract holes from volume and include their exposed walls
Subtract a removed volume that fits inside the original solid · Include newly exposed cavity or tunnel surfaces · Account for entrance and exit openings
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
- Cuboid and cylinder measures
- Compound surface 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
- Subtract a removed volume that fits inside the original solid
- Include newly exposed cavity or tunnel surfaces
- Account for entrance and exit openings
A hole removes material
A cylindrical tunnel of radius r through a cuboid along perpendicular length L removes volume πr²L. Subtract that from the cuboid volume, after checking the circle fits the end face and the tunnel extends through the stated length. Volumes of overlapping holes need an overlap correction.
But its wall becomes exposed
For a straight tunnel open at both ends, remove two circular entrance areas from the old exterior and add the curved tunnel wall: change is −2πr²+2πrL. There are no circular end caps inside an open through-tunnel. A blind hole would have a different surface list, including its bottom.
For a 20 by 12 by 10 cm cuboid with a radius-2 cm tunnel along its 20 cm length, material volume is 2400−80π cm³ and total exposed area is 1120+72π cm².
Sources for this lesson
Pause and explain
Does drilling a through-tunnel always reduce exposed surface area?
Worked example
Find material volume of the 20 by 12 by 10 cm cuboid after the stated radius-2 cm through-tunnel.
Show the worked solution
- Original volume=2400 cm³.
- Tunnel volume=π×2²×20=80π cm³; subtract it.
Answer 2400−80π cm³.
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 the material volume of a 10 cm cube with a radius-1 cm straight cylindrical hole through its full 10 cm thickness.
Give me a hint
Subtract the tunnel from the cube.
Compare my reasoning
- Cube=1000 cm³; hole=π×1²×10=10π cm³.
- Remaining volume=1000−10π cm³.
1000−10π cm³.
Look for these in your work
- Check the hole fits
- Subtract the removed material
For the 20 by 12 by 10 cm cuboid and radius-2 cm through-tunnel along length 20 cm, find total exposed area.
Give me a hint
Start with six cuboid faces; adjust openings and tunnel wall.
Compare my reasoning
- Original area=2(240+200+120)=1120 cm².
- Remove 8π cm² and add 80π cm², giving 1120+72π cm².
1120+72π cm².
Look for these in your work
- Remove two opening disks
- Add the new wall once
For the drilled 10 cm cube above, find exposed area and explain whether it increased or decreased.
Give me a hint
Compare the removed disks with the added curved tunnel.
Compare my reasoning
- Original area=600 cm²; remove 2π cm² and add 20π cm².
- Final area=600+18π cm², an increase despite reduced volume.
600+18π cm²; the exposed area increased.
Look for these in your work
- Distinguish volume from surface
- Justify the direction of change
Common mistakes
- Removing volume can increase exposed surface area.
- An open through-tunnel has no hidden end-cap disks.
What can you explain now?
Does drilling a through-tunnel always reduce exposed surface area?
Compare with the explanation
No; the exposed tunnel wall can exceed the removed entrance areas.
Compare the added curved wall with both removed circular openings.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Does drilling a through-tunnel always reduce exposed surface 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.