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

Will the sofa fit round the corner? The longest object through a corridor

AA SLAA HLAI SLAI HL Solid Also in: Geometry & Voronoi, Pure maths, Optimisation

Research question

What is the longest rod that can be carried horizontally around the corner between two corridors in my home, and what does this imply for moving a sofa of given dimensions?

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

Every house move raises this question. The rod version has a clean calculus answer; the sofa version is a famous unsolved-until-recently problem. Starting from your own corridor measurements makes it personal and practical.

The mathematics you'll need

Trigonometry to express length in terms of an angle
Differentiation to find the minimum of L(θ)
The closed-form result L = (a^(2/3) + b^(2/3))^(3/2)
Extending to objects with width

Where the data comes from

Measure the corridor widths at a corner in your home or school.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

A possible outline

Model the rod touching the inner corner at angle θ.
Write L(θ) and find the minimum.
Derive the general formula and apply to your corridor.
Extend to a rectangular object of width w (numerically).
Discuss tilting, 3-D moves and the “moving sofa problem”.

Pitfalls that cost marks

Confusing the minimum of L(θ) with the longest rod (explain why the minimum is the answer).
Ignoring the object's width without comment.
Not checking the answer with a scale model.

Showing personal engagement

Use measurements from your own home.
Test with a cardboard model.
Research the moving sofa problem and explain Gerver's sofa in your own words.

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	Trigonometry to express length in terms of an angle; Differentiation to find the minimum of L(θ)
AA HL	Good fit	Trigonometry to express length in terms of an angle; Differentiation to find the minimum of L(θ)
AI SL	Good fit	Trigonometry to express length in terms of an angle; Differentiation to find the minimum of L(θ)
AI HL	Good fit	Trigonometry to express length in terms of an angle; Differentiation to find the minimum of L(θ)

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: confusing the minimum of L(θ) with the longest rod (explain why the minimum is the answer); and ignoring the object's width without comment. Say how each one affects your answer.

Use of mathematics (E)

SL: trigonometry to express length in terms of an angle; differentiation to find the minimum of L(θ); the closed-form result L = (a^(2/3) + b^(2/3))^(3/2); extending to objects with width, 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

HL: prove the general formula and investigate the effect of tilting the rod in 3-D.

Extending it for HL

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

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 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?

Trigonometry to express length in terms of an angle, differentiation to find the minimum of L(θ), the closed-form result L = (a^(2/3) + b^(2/3))^(3/2) and extending to objects with width.

Where can I get data for this IA?

Measure the corridor widths at a corner in your home or school. Checked sources and tools: GeoGebra.

Turn this idea into your IA

The planner opens with “Will the sofa fit round the corner? The longest object through a corridor” filled in: the question, the maths and the data sources.

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