Recognise prisms, pyramids and curved solids
Describe solids by their defining structure · Count polygonal faces, edges and vertices · Distinguish a surface from the enclosed region
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
- Plane polygon names
- Length and distance
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
- Describe solids by their defining structure
- Count polygonal faces, edges and vertices
- Distinguish a surface from the enclosed region
Polygonal solids
A polygonal prism has two congruent parallel end polygons and matching connecting faces. For a right prism these side faces are rectangles; an oblique prism need not have rectangular side faces. A pyramid has one polygonal base and triangular faces meeting at one apex.
A cuboid has six rectangular faces, eight vertices and twelve edges. A cube is a cuboid with all edge lengths equal. A triangular prism has two triangle ends and three side faces; a square-based pyramid has one square base and four triangle faces.
Curved boundaries
A right circular cylinder has two circular plane ends and a curved side surface; a right circular cone has one circular base and a curved surface meeting at an apex. A sphere is the surface at a fixed positive distance from its center; the solid ball includes the interior.
Use the question's convention when counting faces of curved solids. Do not silently count a curved surface as a polygonal face. Shape names describe structure, not just the way a perspective drawing looks.
Sources for this lesson
Pause and explain
Does having five faces prove a solid is a triangular prism?
Worked example
Compare the faces of a triangular prism and a square-based pyramid.
Show the worked solution
- The prism has two triangular ends and three connecting faces: five in total.
- The pyramid has one square and four triangles: also five, but with a different arrangement.
Answer Both have five polygonal faces, but they are different solids.
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.
State the number of faces, edges and vertices of a cuboid.
Give me a hint
Count complete structural features, including hidden ones.
Compare my reasoning
- It has six rectangular faces.
- There are twelve edges and eight vertices.
6 faces, 12 edges, 8 vertices.
Look for these in your work
- Include hidden features
- Distinguish faces from edges
Describe the faces and apex arrangement of a square-based pyramid.
Give me a hint
Begin with the base.
Compare my reasoning
- One square base joins four triangular faces.
- The triangles meet at one apex; there are five vertices including the four base corners.
One square and four triangles; one apex; five vertices.
Look for these in your work
- Name the face shapes
- Include the base vertices
A cylinder is drawn like a rectangle with two ellipses. Are its circular ends actually ellipses, and does its side become a plane rectangle on the solid?
Give me a hint
Separate perspective drawing from geometry.
Compare my reasoning
- The ellipses depict circles viewed at an angle.
- The cylinder's side is curved; a rectangle can represent its unrolled side, not a plane face on the closed solid.
The ends are circles; the closed side is curved. The drawing does not change the solid's definition.
Look for these in your work
- Interpret perspective honestly
- Distinguish a net from a closed surface
Common mistakes
- A face count alone may not identify a solid.
- A prism's end polygons are congruent and parallel.
What can you explain now?
Does having five faces prove a solid is a triangular prism?
Compare with the explanation
No; a square-based pyramid also has five faces.
The face shapes and their connections distinguish the solids.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does having five faces prove a solid is a triangular prism?
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.