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

IA idea · Geometry, trigonometry & Voronoi diagrams

Do school catchment areas match the Voronoi diagram of the schools?

AI SLAI HL Solid Also in: Statistics

Research question

How closely do the official catchment areas of the primary schools in [my area] match the Voronoi cells of the schools, and where do they differ most?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Many councils allocate places partly by distance, so catchments should look like Voronoi cells — but roads, rivers and politics get in the way. Comparing the two shows what pure geometry misses.

The mathematics you'll need

Voronoi diagrams from coordinates
Perpendicular bisectors and intersections
Areas of polygons (shoelace formula — explain it)
Percentage overlap between regions

Where the data comes from

Get school coordinates from OpenStreetMap and official catchment maps from your local council.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Real data for this idea

Primary source: OpenStreetMap. Read off coordinates (right-click, “Show address”) or use Export for a small area. Licence: Open Database Licence: credit “© OpenStreetMap contributors”; data you build from it must be shared alike.

OpenStreetMap is under the Open Database Licence: credit “© OpenStreetMap contributors”. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Explain how places are allocated locally.
Build the Voronoi diagram.
Overlay the official catchments.
Measure differences in area and identify causes.
Reflect on fairness and the limits of straight-line distance.

Pitfalls that cost marks

Using schools of different types (e.g., faith schools with different rules).
Not stating the map scale.
Assuming catchments are purely distance-based.

Showing personal engagement

Include your own primary school.
Interview a parent or teacher about admissions.
Suggest where a new school would reduce the largest cell.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Voronoi diagrams from coordinates; Perpendicular bisectors and intersections
AI HL	Good fit	Voronoi diagrams from coordinates; Perpendicular bisectors and intersections

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: using schools of different types (e.g., faith schools with different rules); and not stating the map scale. Say how each one affects your answer.

Use of mathematics (E)

SL: voronoi diagrams from coordinates; perpendicular bisectors and intersections; areas of polygons (shoelace formula — explain it); percentage overlap between regions, each explained in your own words and carried out correctly.

How geometry & voronoi explorations reach the top bands in general: Geometry, trigonometry & Voronoi diagrams IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Weight each school by capacity (a weighted Voronoi diagram) and see whether it matches better.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL geometry & voronoi exploration adds.

See a complete IA, marked

What is the quickest way to deliver a leaflet to every house on my estate? (AI HL) is a complete geometry & voronoi exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Voronoi diagrams from coordinates, perpendicular bisectors and intersections, areas of polygons (shoelace formula — explain it) and percentage overlap between regions.

Where can I get data for this IA?

Get school coordinates from OpenStreetMap and official catchment maps from your local council. Checked sources and tools: OpenStreetMap and GeoGebra.

Turn this idea into your IA

The planner opens with “Do school catchment areas match the Voronoi diagram of the schools?” filled in: the question, the maths and the data sources.

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