Why aren't drinks cans the optimal shape?

AA SLAI SLAA HLAI HL Accessible Also in: Modelling

Research question

How close are real drinks cans to the cylinder with minimum surface area for their volume, and which real-world constraint best explains the difference?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Calculus says the cheapest closed cylinder has height equal to its diameter — yet almost no can looks like that. Explaining the gap, with measurements of real cans, turns a textbook exercise into a genuine investigation.

The mathematics you'll need

Surface area and volume of a cylinder
Differentiation to find a minimum; second-derivative test
Adding constraints (thicker top and bottom, rims) and re-optimising
Percentage difference between real and optimal

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Measure 6–10 real cans and bottles (height, diameter, stated volume); weigh the lid and body material if you can.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Derive the optimal cylinder for a fixed volume.
Measure real cans and compute the percentage extra material each uses.
Add a realistic constraint: the ends are about three times thicker than the wall.
Re-optimise and compare again with real cans.
Discuss grip, stacking, marketing and manufacturing.

Pitfalls that cost marks

Stopping at the basic h = 2r result (every textbook has it).
Using the labelled volume without checking the can is not full to the brim.
Ignoring units when comparing material.

Showing personal engagement

Survey cans from different countries or brands.
Weigh the lid and body to estimate thickness ratios yourself.
Propose a can design and justify it.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Model the domed base or the tapered neck as solids of revolution (HL) and compare total material.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).