Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Why aren't drinks cans the optimal shape?

AA SLAI SLAA HLAI HL Accessible Also in: Modelling, Optimisation

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?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

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

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.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Surface area and volume of a cylinder; Differentiation to find a minimum; second-derivative test
AA HL	Fits, but add an HL technique	Surface area and volume of a cylinder; Differentiation to find a minimum; second-derivative test
AI SL	Good fit	Surface area and volume of a cylinder; Differentiation to find a minimum; second-derivative test
AI HL	Fits, but add an HL technique	Surface area and volume of a cylinder; Differentiation to find a minimum; second-derivative test

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: stopping at the basic h = 2r result (every textbook has it); and using the labelled volume without checking the can is not full to the brim. Say how each one affects your answer.

Use of mathematics (E)

SL: 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, each explained in your own words and carried out correctly.

How calculus explorations reach the top bands in general: Calculus & optimisation IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

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

Extending it for HL

Add an HL technique the question genuinely needs: what an HL calculus exploration adds.

See a complete IA, marked

Drinks-can optimisation: a mid-band AA SL draft, annotated to show how to improve (AA SL) explores a question like this one, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AI SL, AA HL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

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 and percentage difference between real and optimal.

Where can I get data for this IA?

Measure 6–10 real cans and bottles (height, diameter, stated volume); weigh the lid and body material if you can. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Why aren't drinks cans the optimal shape?” filled in: the question, the maths and the data sources.

Plan this IA free

Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in

Also: free IA self-check · A–E checklist (PDF) · have a draft? AI feedback or a review with Pete (€100)

Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · IA modelling, step by step