Calculate surface area and volume of solids
Choose a solid formula, interpret a prism by its uniform cross-section, and keep surface area and volume units separate.
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
Which unit is appropriate for volume?
What you will learn
- Calculate surface area and volume of common solids
- Use formulas supplied by the syllabus where indicated
- Find missing dimensions from a volume or area relationship
Separate the outside from the space inside
Surface area covers exposed faces; volume measures the space occupied. A cuboid has volume lwh and total surface area 2(lw + lh + wh). A prism's volume is cross-sectional area × length.
A prism means any solid with a uniform cross-section. Cylinder volume is πr²h; total surface area adds two circular ends to the curved surface.
Know which formula is supplied
Cambridge gives curved surface area of cylinders and cones, sphere surface area, and volumes of prisms, pyramids, cylinders, cones and spheres. Other formulas, such as a cuboid's total surface area, need to be understood or constructed from faces.
Answers may be left in terms of π. State units as square units for area and cubic units for volume; use the slant height for a cone's curved surface area.
Pause and explain
What does a prism's volume equal?
Worked example
A cylinder has radius 3 cm and height 5 cm. Find its volume and total surface area.
Show the worked solution
- Volume = πr²h = π(3²)(5) = 45π cm³.
- Curved surface area = 2πrh = 30π cm².
- Add the two circular ends: 30π + 2π(3²) = 48π cm².
Answer Volume 45π cm³; total surface area 48π 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 rectangular prism measures 4 cm by 3 cm by 8 cm. Find its volume.
Give me a hint
Multiply the area of a rectangular end by the prism length.
Compare my reasoning
- Cross-sectional area = 4 × 3 = 12 cm².
- Volume = 12 × 8.
- Volume = 96 cm³.
96 cm³.
Look for these in your work
- Cross-section selected
- Length multiplied once
- Cubic units stated
A sphere has radius 3 cm. Find its exact volume and surface area.
Give me a hint
Use the supplied formulas 4/3πr³ and 4πr².
Compare my reasoning
- Volume = 4/3 × π × 27 = 36π cm³.
- Surface area = 4π × 9.
- Surface area = 36π cm².
Volume 36π cm³; surface area 36π cm².
Look for these in your work
- Radius cubed for volume
- Radius squared for area
- Units distinguished
A closed cylindrical container has radius 4 cm and height 10 cm. Find its capacity in cm³ and litres, and its total outer surface area.
Give me a hint
Calculate cylinder volume; total area includes the curved side and both circular ends.
Compare my reasoning
- Volume = π(4²)(10) = 160π cm³.
- Capacity in litres = 160π/1000 ≈ 0.503 litres.
- Total surface area = 2π(4)(10) + 2π(4²) = 112π cm².
Capacity 160π cm³ ≈ 0.503 L; total area 112π cm².
Look for these in your work
- Volume formula applied
- cm³ converted to litres
- Both end faces included
Common mistakes
- Confusing slant height with vertical height in cone area
- Forgetting end faces in total surface area
- Reporting volume in square units
- Using the area formula where a volume formula is needed
What can you explain now?
A prism has cross-sectional area 12 cm² and length 8 cm. Find its volume.
Compare with the explanation
96 cm³
Volume = cross-sectional area × length.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do dimensions and exposed faces determine a solid's volume and 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.