Mathematics 7–10 · Year 9
Surface area of a cylinder: peeling the label off a tin
Measurement and space
The idea
The curved surface of a cylinder unrolls into a rectangle whose length is the circumference and whose height is the cylinder's height, so the surface area is 2πr² + 2πrh.
What you need
- Unopened tinned food cans with paper labels, one per pair (the worked figures use a can of radius 3.7 cm and height 11.0 cm; each pair measures its own)
- Ruler, scissors, 1 cm grid paper
- Calculator
How to do it
- Measure the diameter and height of the can; compute the circumference 2πr and predict the label's length before removing it.
- Peel the label off, lay it flat and measure its length and height; compare the length with the predicted circumference.
- Place the can on grid paper and trace an end; estimate its area by counting squares, then compute πr².
- Add the label area and two end areas to get the total surface area; compare with the formula 2πr² + 2πrh.
- Compute what fraction of the surface area is the curved part for this can, then for a can twice as tall.
What you should see
For r = 3.7 cm and h = 11.0 cm the circumference is 23.25 cm, and the flattened label is slightly longer where its ends overlap at the glued seam; the curved area is 255.7 cm², each end is 43.0 cm², and the total is 341.7 cm². A traced end centred on a grid corner has 32 whole squares inside and 60 squares touched, so its area lies between 32 and 60 cm²; counting part squares as halves gives 46 cm² against the computed 43.0 cm², so at this size the grid only brackets the answer. The curved part is h ÷ (r + h) of the total, 75 per cent for this can and 86 per cent for a can twice as tall. The learner knows it worked when the measured label length matches 2πr once the overlap is allowed for and the total agrees with the formula within 3 per cent.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The curved surface area is πr²h.
- The label's length is the diameter.
- Surface area and volume grow in the same proportion.
Safety card
Hazards
- sharp rim on an opened can
Controls
- use unopened cans
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Area and surface area A; page read 2026-09-22MA5-ARE-C-01
- Australian Curriculum v9AC9M9M01
Sources
The pages the author read to write this activity.