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

Is a hanging chain a parabola? Catenary versus quadratic models

AA SLAA HLAI SLAI HL Solid Also in: Calculus, Art & music

Research question

How much better does a catenary y = a cosh(x/a) + c describe a hanging chain than the best-fitting parabola, and does the difference matter for a real design?

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

Galileo thought a hanging chain formed a parabola; it does not, but the two curves are so close that the difference is surprisingly hard to see. Measuring a real chain and quantifying the gap is a lovely way to test a famous claim with your own data.

The mathematics you'll need

Quadratic models through three points and by least squares (SL)
Exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it
Sum of squared residuals to compare models
Derivatives to find the angle the chain makes at its supports
HL: arc length by integration, or solving for the parameter numerically

Modelling the data, step by step

This idea usually needs quadratic or exponential models. See it worked step by step, with a criterion tip at every step: Quadratic: three points, completing the square, regression · Exponential y = a e^(kx) + c and y = a·bˣ · Choosing and comparing models.

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

Hang a light chain or necklace against a grid (or photograph it square-on with a ruler in shot), read 15–25 points off the photo in GeoGebra, and repeat for two different span-to-sag ratios.

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

Photograph the chain square-on and set up axes with the lowest point at the origin; state your scale.
Fit a parabola by least squares; record the residuals.
Introduce the catenary, explain where cosh comes from, and choose a by trial or technology.
Compare the models with residual plots and sums of squares; repeat for a deeper sag where the curves diverge more.
Discuss where the difference matters (arches, cables) and where it does not.

Pitfalls that cost marks

Photographing at an angle — perspective distorts the shape more than the difference you are trying to measure.
Claiming the catenary is “better” from one number without looking at where the residuals are.
Using cosh without explaining what it is; unfamiliar maths must be explained.

Showing personal engagement

Predict before measuring which model will win and by how much.
Link it to a structure you know (a local bridge, the Gateway Arch, a washing line) and check it.
Explain why the difference grows as the chain sags more — and test that prediction.

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	Quadratic models through three points and by least squares; Exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it
AA HL	Good fit	Quadratic models through three points and by least squares; Exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it
AI SL	Good fit	Quadratic models through three points and by least squares; Exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it
AI HL	Good fit	Quadratic models through three points and by least squares; Exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it

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: photographing at an angle — perspective distorts the shape more than the difference you are trying to measure; and claiming the catenary is “better” from one number without looking at where the residuals are. Say how each one affects your answer.

Use of mathematics (E)

SL: quadratic models through three points and by least squares; exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it; sum of squared residuals to compare models; derivatives to find the angle the chain makes at its supports, each explained in your own words and carried out correctly.

HL adds: arc length by integration, or solving for the parameter numerically, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

HL students can derive the catenary from a force balance (a differential equation) or compute the chain's length by integration and compare it with the measured length.

Extending it for HL

This idea already has HL mathematics in it: arc length by integration, or solving for the parameter numerically. 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) 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, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Quadratic models through three points and by least squares (SL), exponential functions and the definition cosh x = (eˣ + e⁻ˣ)/2 — new, explain it, sum of squared residuals to compare models, derivatives to find the angle the chain makes at its supports and hL: arc length by integration, or solving for the parameter numerically.

Where can I get data for this IA?

Hang a light chain or necklace against a grid (or photograph it square-on with a ruler in shot), read 15–25 points off the photo in GeoGebra, and repeat for two different span-to-sag ratios. Checked sources and tools: GeoGebra and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Is a hanging chain a parabola? Catenary versus quadratic models” filled in: the question, the maths and the data sources.

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