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

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?

Adapt it: change the place, the data or the comparison until the question is yours.

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

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

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.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

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.

Taking it further

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

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).