Solve problems with compound shapes and solids
Split or subtract familiar shapes and solids, account for hidden joins, and find perimeters, areas, surface areas and volumes.
Section 6 · Mensuration
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
When finding the outside area of joined solids, what happens to the shared face?
What you will learn
- Decompose compound plane shapes and parts
- Combine or subtract volumes and exposed surface areas
- Check shared boundaries and exact π forms
Split the shape into familiar pieces
For compound area, add non-overlapping pieces or subtract a cut-out from a larger shape. For perimeter, trace only the outside boundary; an internal join is not part of the perimeter.
A part of a shape may be a sector, semicircle or other fraction. Keep each piece's units consistent and include only the boundary requested.
Hidden joins do not remain exposed
For a compound solid, add component volumes when they do not overlap. For external surface area, include only faces exposed to the outside; do not count a joined circular base twice.
C6.5 and E6.5 include parts and compound shapes or solids; Extended examples include a frustum. Exact answers in terms of π may be requested.
Pause and explain
Two hemispheres joined along their flat circular faces make which solid?
Worked example
A capsule is a cylinder of radius 3 cm and length 5 cm between two hemispherical ends. Find its volume and external surface area.
Show the worked solution
- The two hemispheres make one sphere: volume = 4/3π(3³) = 36π cm³.
- Cylinder volume = π(3²)(5) = 45π cm³; total volume = 81π cm³.
- External area = cylinder curved area 2π(3)(5) plus sphere area 4π(3²) = 30π + 36π = 66π cm².
Answer Volume 81π cm³; external surface area 66π cm².
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 shape consists of a 10 cm by 6 cm rectangle with a 2 cm radius semicircle attached along its 4 cm diameter. Find its area.
Give me a hint
Add the rectangle and half-circle areas; the shared diameter is not an area overlap.
Compare my reasoning
- Rectangle area = 10 × 6 = 60 cm².
- Semicircle area = 1/2 × π × 2² = 2π cm².
- Total area = 60 + 2π cm².
60 + 2π cm².
Look for these in your work
- Pieces identified
- Semicircle fraction used
- Exact π form retained
A solid is made from a cylinder of radius 2 cm and height 7 cm with a hemisphere of radius 2 cm on top. Find its volume.
Give me a hint
Add cylinder volume and half a sphere; the joined circular face is internal.
Compare my reasoning
- Cylinder volume = π(2²)(7) = 28π cm³.
- Hemisphere volume = 1/2 × 4/3π(2³) = 16π/3 cm³.
- Total = 28π + 16π/3 = 100π/3 cm³.
100π/3 cm³.
Look for these in your work
- Components separated
- Hemisphere is half a sphere
- Fractions combined
A cylindrical pipe has outside radius 5 cm, inside radius 4 cm and length 2 m. Find the volume of metal in cm³.
Give me a hint
Convert the length first, then subtract the hollow volume from the outside cylinder.
Compare my reasoning
- 2 m = 200 cm.
- Metal volume = π(5²)(200) − π(4²)(200).
- This is π(25 − 16)(200) = 1800π cm³.
1800π cm³.
Look for these in your work
- Length converted to centimetres
- Hollow core subtracted
- Exact answer retained
Common mistakes
- Double-counting a shared face
- Adding an internal boundary to the outside perimeter
- Subtracting overlapping pieces incorrectly
- Using a hemisphere as a full sphere
What can you explain now?
Two non-overlapping solids have volumes 12 cm³ and 20 cm³. What is their combined volume?
Compare with the explanation
32 cm³
Add the volumes when the solids do not overlap.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do you decide which component areas, boundaries and volumes to add or remove in a compound figure?
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.