Identify a solid cross section from the cutting plane
Recognise sections parallel to bases or through axes · Distinguish a section from a view or silhouette · Handle a sphere cut away from its center
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
- Prisms, cones and spheres
- Right triangles
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
- Recognise sections parallel to bases or through axes
- Distinguish a section from a view or silhouette
- Handle a sphere cut away from its center
The plane matters
A cross section is the region where a plane cuts a solid. It is not necessarily the shape of the whole object viewed from the outside. A prism cut parallel to its ends reproduces its end shape. A right cylinder cut parallel to its circular bases gives a circle; a cut through its axis gives a rectangle. A right cone cut parallel to its base gives a circle, while a plane through its axis gives a triangle.
Position matters too
A plane cutting a sphere of radius R at perpendicular distance d from its center gives a circle of radius √(R²−d²) when |d|<R. At |d|=R it touches at a single point; beyond the sphere there is no section. A central cut is the largest circle.
An oblique cube section is not forced to be a square: a plane meeting six faces in six distinct non-zero segments forms a hexagon. Identify the actual plane and intersections before naming the section.
Sources for this lesson
Pause and explain
A cylinder is cut by a plane containing its axis. What is the section of a right cylinder?
Worked example
A radius-5 cm sphere is cut by a plane 3 cm from its center. Find the section radius.
Show the worked solution
- In the perpendicular right triangle, section radius²=5²−3²=16.
- The non-negative section radius is 4 cm.
Answer 4 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.
Name the sections of a right cylinder cut parallel to its base and through its axis.
Give me a hint
Compare the two cutting directions.
Compare my reasoning
- A base-parallel section is a circle.
- An axial section is a rectangle.
Circle; rectangle.
Look for these in your work
- Use the plane direction
- Distinguish the two cases
For a sphere of radius 5 cm, describe cuts at center distances 3 cm, 5 cm and 6 cm.
Give me a hint
Compare the distance with the radius.
Compare my reasoning
- At 3 cm the circle radius is √(25−9)=4 cm.
- At 5 cm there is a tangent point; at 6 cm there is no intersection.
Radius-4 cm circle; one point; no section.
Look for these in your work
- Check intersection conditions
- Take a non-negative length
A plane meets each of the six faces of a cube in a distinct non-zero segment, with no whole face in the plane. Must its section be a square? Explain.
Give me a hint
Count the resulting boundary segments.
Compare my reasoning
- Six face segments give six sides of the section polygon.
- It is a hexagon, so the section is not forced to be square.
No; the six segments form a hexagonal section.
Look for these in your work
- Use the supplied intersections
- Avoid a solid-name-only assumption
Common mistakes
- A silhouette is not a specified plane section.
- An off-center sphere section is smaller than the central one.
What can you explain now?
A cylinder is cut by a plane containing its axis. What is the section of a right cylinder?
Compare with the explanation
A rectangle.
The plane contains the cylinder height and a base diameter.
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Make it stick
A cylinder is cut by a plane containing its axis. What is the section of a right cylinder?
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.