Learn: a label, a lid or the whole container?
A paper label wraps around the curved side of a can. A lid covers one circular end. To find how much material is needed, first decide which surfaces count. Surface area uses square units such as cm²; volume uses cubic units such as cm³.
This page uses a right circular cylinder: two equal circular bases with straight sides perpendicular to the bases. The radius r runs from the centre of a base to its edge, and the height h measures the perpendicular distance between bases. π (pi) is approximately 3.14159.
Unwrap the curved side
Cut a paper label along a vertical line and flatten it. The result is a rectangle: its width is the circle’s circumference, 2πr, and its height is h. Rectangle area is width × height, giving curved-side area = 2πrh. Each circular end adds πr².
| Surfaces included | Area | Example task |
|---|---|---|
| Curved side only | 2πrh | A label around the side |
| Side and one end | 2πrh + πr² | An ideal open-top container |
| Side and both ends | 2πrh + 2πr² | Covering a closed cylinder completely |
Worked example: r = 3 cm, h = 5 cm
- Side: 2π × 3 × 5 = 30π ≈ 94.25 cm².
- One circular end: π × 3² = 9π cm².
- Closed total: 30π + 2(9π) = 48π ≈ 150.80 cm².
- Interpret: the label needs the side area; a closed model needs both ends too. Real packaging may require overlap or allowances, which this geometry does not include.
Explore: change one dimension
Predict first: what happens when you double the radius while keeping the height fixed? Compare that with doubling only the height.
Curved side: 60π ≈ 188.50 cm². Closed total: 78π ≈ 245.04 cm².
The sliders use whole centimetres. Volume measures the space inside; surface area measures the material covering the outside. The written examples below work without this explorer.
More worked comparisons
A roller: r = 0.5 m, h = 2 m
Curved side: 2π(0.5)(2) = 2π ≈ 6.28 m². Two ends: 2π(0.5)² = 0.5π m². Closed total: 2.5π ≈ 7.85 m². Every term uses square metres; adding cm² to m² without conversion would be incorrect.
A canister: r = 7 cm, h = 12 cm
The side is 2π(7)(12) = 168π cm². The two ends add 2π(7²) = 98π cm². Total = 266π ≈ 835.66 cm². A label covering only the side needs 168π ≈ 527.79 cm² before any overlap allowance.
Connect back to volume
A larger radius changes side area and end area differently: doubling r doubles 2πrh but quadruples 2πr². At fixed height, the closed total therefore does not simply double or quadruple. By comparison, the volume quadruples.
Practice: choose the surfaces first
1. An open-top cylinder has radius 2 cm and height 6 cm. Find its area.
Include side plus one base: 2π(2)(6) + π(2²) = 24π + 4π = 28π ≈ 87.96 cm².
2. A closed cylinder has diameter 10 cm and height 8 cm. Find its total area.
r = 10 ÷ 2 = 5 cm. Side area = 2π(5)(8) = 80π cm²; two ends = 2π(5²) = 50π cm². Total = 130π ≈ 408.41 cm².
3. For that cylinder, how much area belongs to a label covering only the side?
Use only 80π ≈ 251.33 cm². Adding circular ends would answer a different question.
4. Why doesn’t doubling the height double the closed total surface area?
The side area doubles, but the two circular ends are unchanged. The new total is 4πrh + 2πr², not twice the original total.
Review before you choose a formula
- Sketch the cylinder and mark the surfaces that count.
- Convert diameter to radius and use one length unit.
- Keep square units on every area term and round at the end.
Explain the rectangle-and-circles reasoning to someone else, then revisit a different example later. This short practice is a starting point for checking understanding.
Keep learning
Compare cylinder volume and surface area or explore geometry topics and tutoring.
Optional reference: OpenStax: volume and surface area. This page provides its own explanations and practice.