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

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?

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

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)

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.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Area of a trapezium in terms of an angle; Comparing optima across shapes
AA HL	Fits, but add an HL technique	Area of a trapezium in terms of an angle; Comparing optima across shapes
AI SL	Good fit	Area of a trapezium in terms of an angle; Differentiation of trigonometric functions (AA) or technology to find the maximum
AI HL	Fits, but add an HL technique	Area of a trapezium in terms of an angle; Differentiation of trigonometric functions (AA) or technology to find the maximum

Level: Accessible. A good first extended piece of maths, with room to go deeper. 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: optimising one variable when two can vary; and not checking the endpoints of the domain. Say how each one affects your answer.

Use of mathematics (E)

SL: area of a trapezium in terms of an angle; comparing optima across shapes; flow rate proportional to area (a modelling assumption); differentiation of trigonometric functions (AA) or technology to find the maximum, 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: use partial reasoning (fix one variable, optimise the other) or Lagrange-style thinking to handle both variables at once.

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, AI SL, AA HL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

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 and flow rate proportional to area (a modelling assumption).

Where can I get data for this IA?

Measure gutters on your home or school; photograph their cross-sections with a scale. Checked sources and tools: GeoGebra.

Turn this idea into your IA

The planner opens with “What is the best shape for a rain gutter?” filled in: the question, the maths and the data sources.

Plan this IA free

Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in

Also: free IA self-check · A–E checklist (PDF) · have a draft? AI feedback or a review with Pete (€100)

Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · IA modelling, step by step