Identify the region and axis before generating a solid
Generate a cylinder, cone or sphere from a specified region · Use the rotating region distance from its axis as radius · Show that a different axis can generate a different solid
Section Paper 4 · Historical Extended · calculator
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, cone and sphere volume
- Plane shapes and axes
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
- Generate a cylinder, cone or sphere from a specified region
- Use the rotating region distance from its axis as radius
- Show that a different axis can generate a different solid
Rotate a filled region
Rotating a rectangle about one of its sides generates a cylinder: the axis-side length gives height and the perpendicular width gives radius. Rotating a right triangle about one of its perpendicular legs generates a cone. Rotating a semicircular region about its diameter generates a sphere. These are filled regions rotating through a full turn, not only boundary lines.
State the axis
The diagram uses rectangle width 3 and height 4 about its vertical side, a right triangle with radial leg 3 and axial leg 4, and a radius-3 semicircular region about its vertical diameter. Their volumes are 36π, 12π and 36π cubic units respectively.
Rotating the 3 by 4 rectangle about its other side instead gives radius 4 and height 3, with volume 48π. If a region does not reach the axis, it may generate a hollow solid rather than the filled solid assumed by a simple formula.
Sources for this lesson
Pause and explain
A right triangle rotates about one perpendicular leg. Which leg supplies the cone radius?
Worked example
Rotate a 3 cm wide, 4 cm high rectangle about its 4 cm side. Name the solid and find its volume.
Show the worked solution
- The solid is a cylinder with radius 3 cm and height 4 cm.
- Volume=π×9×4=36π cm³.
Answer Cylinder; 36π 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.
A rectangle 3 cm wide and 4 cm high rotates about its 4 cm side. Find the cylinder radius, height and volume.
Give me a hint
Use the width as the radius.
Compare my reasoning
- r=3 cm and h=4 cm.
- Volume=π×3²×4=36π cm³.
Radius 3 cm; height 4 cm; volume 36π cm³.
Look for these in your work
- Identify the rotation axis
- Read perpendicular distance as radius
A right triangle with radial leg 3 cm and axial leg 4 cm rotates about the axial leg. A radius-3 cm semicircular region rotates about its diameter. Find the generated volumes.
Give me a hint
Use cone and sphere formulae respectively.
Compare my reasoning
- Cone volume=(1/3)π×9×4=12π cm³.
- Sphere volume=(4/3)π×27=36π cm³.
Cone 12π cm³; sphere 36π cm³.
Look for these in your work
- Name each generated solid
- Use its correct volume rule
Compare the volumes when a 3 cm by 4 cm rectangle rotates about its 4 cm side and about its 3 cm side.
Give me a hint
The radius and height exchange roles.
Compare my reasoning
- About the 4 cm side: π×3²×4=36π cm³.
- About the 3 cm side: π×4²×3=48π cm³.
36π cm³ and 48π cm³; the different axes give different volumes.
Look for these in your work
- State the axis for each case
- Do not treat radius and height symmetrically
Common mistakes
- The side on the axis determines height, not radius.
- A different axis can change the generated volume.
What can you explain now?
A right triangle rotates about one perpendicular leg. Which leg supplies the cone radius?
Compare with the explanation
The perpendicular leg away from the axis.
Its distance from the axis sweeps out the circular base.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
A right triangle rotates about one perpendicular leg. Which leg supplies the cone radius?
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.