Calculate sphere area and volume with dimensional checks
Use 4πr² for sphere surface area · Use (4/3)πr³ for sphere volume · Check scaling and distinguish a sphere from a disk
Section E6.4 · Extended · Surface areas and volumes
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
- Circle radius and diameter
- Powers and cubic units
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
- Use 4πr² for sphere surface area
- Use (4/3)πr³ for sphere volume
- Check scaling and distinguish a sphere from a disk
Surface and interior
A sphere of radius r has surface area 4πr² and volume (4/3)πr³. Its circular silhouette has area πr², a different quantity from the complete curved surface. If diameter is supplied, halve it first. Use the given formula sheet rather than guessing from the appearance of a circle.
Check dimensions and scaling
Area uses r² and square units, while volume uses r³ and cubic units. Doubling radius multiplies sphere area by four and volume by eight. A measured ball may be modelled as a sphere, but that idealisation does not establish that its material is uniform or its surface perfectly spherical.
Sources for this lesson
Pause and explain
By what factor does sphere volume change when radius doubles?
Worked example
Find surface area and volume of a radius-3 cm sphere.
Show the worked solution
- Surface area=4π×9=36π cm².
- Volume=(4/3)π×27=36π cm³.
Answer 36π cm²; 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.
Find surface area and volume of a sphere of radius 2 cm.
Give me a hint
Use the square for area and cube for volume.
Compare my reasoning
- Area=4π×4=16π cm².
- Volume=(4/3)π×8=(32/3)π cm³.
16π cm²; (32/3)π cm³.
Look for these in your work
- Keep the formulae distinct
- Use the correct dimensions
A sphere has diameter 12 m. Find surface area and volume.
Give me a hint
Its radius is 6 m.
Compare my reasoning
- Area=4π×36=144π m².
- Volume=(4/3)π×216=288π m³.
144π m²; 288π m³.
Look for these in your work
- Halve the diameter
- Retain exact pi
Two spherical balls have radii 2 cm and 4 cm. Compare their surface areas and volumes as ratios smaller:larger.
Give me a hint
Use powers of the radius ratio 1:2.
Compare my reasoning
- Area ratio=1²:2²=1:4.
- Volume ratio=1³:2³=1:8.
Surface area 1:4; volume 1:8.
Look for these in your work
- Use a dimensional scale
- State ratio direction
Common mistakes
- A circular silhouette is not the whole sphere surface.
- Numerically equal area and volume expressions still have different units.
What can you explain now?
By what factor does sphere volume change when radius doubles?
Compare with the explanation
8.
Volume depends on the cube of radius.
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Make it stick
By what factor does sphere volume change when radius doubles?
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.