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

What is the volume of an egg?

AA HLAI HLAA SL Solid Also in: Modelling

Research question

How accurately can the volume of a hen's egg be found by modelling its profile with a function and using a volume of revolution, compared with measuring it by water displacement?

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Why it makes a good exploration

An egg is not an ellipsoid, and mathematicians have published several “egg equations”. Photographing an egg, fitting its profile and integrating to get the volume — then checking with a measuring jug — gives a result you can verify at home.

The mathematics you'll need

Fitting a profile curve (polynomial, ellipse, or a published egg formula)
Volume of revolution V = π∫y² dx (HL)
Numerical integration or technology
Percentage error against displacement

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 5 eggs side-on on a grid; measure each volume by water displacement in a measuring cylinder.

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

Explain the idea of a solid of revolution.
Extract profile points and fit two different models.
Integrate to find volumes; compare with displacement.
Test the simple formula V ≈ (π/6)LB² against your results.
Reflect on measurement error in displacement and in the photo.

Pitfalls that cost marks

Rotating about the wrong axis.
Only fitting half the egg without checking symmetry.
Not estimating the uncertainty in the displacement measurement.

Showing personal engagement

Compare egg sizes (S, M, L) with the sizes' official mass ranges.
Try a quail or duck egg.
Design a better approximate formula from your data.

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	Fitting a profile curve (polynomial, ellipse, or a published egg formula); Numerical integration or technology
AA HL	Good fit	Fitting a profile curve (polynomial, ellipse, or a published egg formula); Volume of revolution V = π∫y² dx
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 profile curve (polynomial, ellipse, or a published egg formula); Volume of revolution V = π∫y² dx

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: rotating about the wrong axis; and only fitting half the egg without checking symmetry. Say how each one affects your answer.

Use of mathematics (E)

SL: fitting a profile curve (polynomial, ellipse, or a published egg formula); numerical integration or technology; percentage error against displacement, each explained in your own words and carried out correctly.

HL adds: volume of revolution V = π∫y² dx, 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

SL students can estimate the volume by summing discs (a Riemann-style approximation) and compare with the exact integral.

Extending it for HL

This idea already has HL mathematics in it: Volume of revolution V = π∫y² dx. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete calculus exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA HL, AI HL and AA SL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Fitting a profile curve (polynomial, ellipse, or a published egg formula), volume of revolution V = π∫y² dx (HL), numerical integration or technology and percentage error against displacement.

Where can I get data for this IA?

Photograph 5 eggs side-on on a grid; measure each volume by water displacement in a measuring cylinder. Checked sources and tools: GeoGebra and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “What is the volume of an egg?” filled in: the question, the maths and the data sources.

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