Cylinder volume and surface area
Use the circular cross-section for capacity and unroll the curved surface for its area.
Section Paper 3 · Core · Calculator
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Find circle area and circumference.
- Interpret square and cubic units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Unrolling a cylinder's curved surface gives what width?
What you will learn
- Find cylinder volume, curved area and total surface area.
- Solve inverse dimensions and open-cylinder problems.
Volume stacks circular discs
A cylinder of radius r and perpendicular height h has volume πr²h. For r = 3 cm and h = 10 cm this is 90π cm³. If a diameter is supplied, halve it before squaring. To recover a height, divide volume by πr²; to recover a radius, take √(V/(πh)).
A tank's inside dimensions determine capacity. Outer measurements include wall thickness and do not automatically give the internal volume. When thickness is ignored explicitly, the ideal cylinder is a suitable model. Convert cubic volume into litres only after applying consistent length units.
Unroll the curved wall
A cylinder's curved surface unrolls into a rectangle of width equal to circumference 2πr and height h. Its area is 2πrh. A closed cylinder has two circular ends, giving total area 2πrh + 2πr². An open-top cylinder has one base, so area is 2πrh + πr².
For r = 3 and h = 10 cm, curved area is 60π cm², closed area 78π cm² and open-top area 69π cm². State which boundary the question asks for. A label around a can needs curved area alone; material for a can with two ends requires total area.
Pause and explain
An open-top cylinder has radius 2 cm and height 5 cm. Find its material area.
Worked example
A closed cylinder has radius 3 cm and height 10 cm. Find exact volume and total area.
Show the worked solution
- Volume is π × 3² × 10 = 90π cm³.
- Curved area is 2π × 3 × 10 = 60π cm².
- Two ends add 2π × 3² = 18π cm², giving total area 78π cm².
Answer 90π cm³; 78π cm².
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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 volume and curved area for a cylinder of radius 4 cm and height 6 cm.
Give me a hint
The radius is squared for volume but not for the curved strip.
Compare my reasoning
- Volume is π × 16 × 6 = 96π cm³.
- Curved area is 2π × 4 × 6 = 48π cm².
- No circular ends belong in curved area alone.
96π cm³; 48π cm².
Look for these in your work
- Used the correct power of radius.
- Distinguished curved from total area.
A cylinder has volume 180π cm³ and radius 3 cm. Find its height.
Give me a hint
Divide by the circular cross-sectional area.
Compare my reasoning
- The cross-section is 9π cm².
- Height is 180π/(9π) = 20 cm.
- Check π × 9 × 20 = 180π cm³.
20 cm.
Look for these in your work
- Cancelled pi consistently.
- Verified the recovered dimension.
A cylindrical label covers only the curved wall of a can of diameter 8 cm and height 12 cm. Find exact label area and describe what closed-can total area would add.
Give me a hint
Halve diameter and count only the requested surfaces.
Compare my reasoning
- Radius is 4 cm, giving curved area 2π × 4 × 12 = 96π cm².
- A closed can also has two ends totalling 32π cm².
- Its total would be 128π cm², while the label stays 96π cm².
Label 96π cm²; closed total 128π cm².
Look for these in your work
- Converted diameter to radius.
- Kept the label's boundary separate from the whole can.
Common mistakes
- Using diameter as radius.
- Including both lids for an open cylinder.
- Using outside measurements for capacity without considering thickness.
What can you explain now?
Which radius power appears in cylinder volume?
Compare with the explanation
r²
A cylinder volume is circular cross-sectional area πr² multiplied by height.
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Make it stick
Tomorrow, explain cylinder area using its net before using a formula.
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.