Cylinder volume and surface area
Use the circular cross-section for capacity and unroll the curved surface for its area.
Core and Extended.
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.
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.
After trying it yourself, choose your next review. This is your self-assessment.
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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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources