What is the best shape for a rain gutter?

AA SLAI SLAA HLAI HL Accessible Also in: Geometry & Voronoi

Research question

For a strip of metal of fixed width, which fold angle and shape (trapezium, rectangle, semicircle) gives the gutter with the greatest cross-sectional area, and how do real gutters compare?

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

Why it makes a good exploration

Folding a flat sheet into a channel is a real manufacturing decision. The optimisation combines trigonometry and calculus, and the comparison with gutters on your own house brings in practical constraints.

The mathematics you'll need

Area of a trapezium in terms of an angle
Differentiation of trigonometric functions (AA) or technology to find the maximum (AI)
Comparing optima across shapes
Flow rate proportional to area (a modelling assumption)

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 gutters on your home or school; photograph their cross-sections with a scale.

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 a trapezoidal gutter folded from a strip of width w.
Maximise area with respect to fold angle and side length.
Compare with rectangular and semicircular shapes.
Measure real gutters and compute how close they are to optimal.
Reflect: debris, strength, brackets and appearance.

Pitfalls that cost marks

Optimising one variable when two can vary.
Not checking the endpoints of the domain.
Assuming maximum area means maximum flow without saying so.

Showing personal engagement

Link it to flooding or a heavy rain event where you live.
Build a card model and test capacity with water.
Discuss why the theoretical best is rarely built.

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

Taking it further

HL: use partial reasoning (fix one variable, optimise the other) or Lagrange-style thinking to handle both variables at once.

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).