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

Where should the 125 ml line go on a glass?

AA HLAI HL Solid Also in: Modelling

Research question

Using a volume of revolution of a fitted profile, at what height should the 125 ml and 175 ml lines be marked on a particular glass, and how accurate is the prediction?

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

Measures in bars and restaurants must be accurate by law in many countries. Modelling a real glass and predicting the height of a fill line gives an answer you can test with a measuring jug.

The mathematics you'll need

Fitting a polynomial or other function to the glass profile
Volume of revolution about the y-axis
Solving V(h) = 125 for h numerically
Error analysis

Modelling the data, step by step

This idea usually needs a polynomial model. See it worked step by step, with a criterion tip at every step: Cubic and polynomials (and overfitting).

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Photograph a stemmed glass with a ruler, extract profile points; test fill lines with a measuring syringe or jug.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Fit the inside profile x = f(y).
Set up V(h) = π∫₀ʰ x² dy.
Solve for heights giving 125 ml and 175 ml.
Test by filling and measuring; compare.
Discuss glass thickness and meniscus.

Pitfalls that cost marks

Modelling the outside rather than the inside of the glass.
Rotating about the wrong axis.
Forgetting the curved base.

Showing personal engagement

Choose a glass from home with personal significance.
Compare two glass shapes and discuss which makes pouring errors more obvious.
Link to consumer law on serving sizes.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Fitting a polynomial or other function to the glass profile; Volume of revolution about the y-axis
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Fitting a polynomial or other function to the glass profile; Volume of revolution about the y-axis

Level: Solid. Needs some independent work beyond class examples. 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: modelling the outside rather than the inside of the glass; and rotating about the wrong axis. Say how each one affects your answer.

Use of mathematics (E)

HL: fitting a polynomial or other function to the glass profile; volume of revolution about the y-axis; solving V(h) = 125 for h numerically; error analysis, 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

Find the rate at which the liquid level rises when poured at a constant rate (related rates) and explain why it is slow at the bowl's widest point.

Extending it for HL

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

See a complete IA, marked

Where should the 125 ml line go on a wine glass? A mid-band AA HL draft, annotated to show how to improve (AA HL) 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 HL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Fitting a polynomial or other function to the glass profile, volume of revolution about the y-axis, solving V(h) = 125 for h numerically and error analysis.

Where can I get data for this IA?

Photograph a stemmed glass with a ruler, extract profile points; test fill lines with a measuring syringe or jug. Checked sources and tools: GeoGebra and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Where should the 125 ml line go on a glass?” 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