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

IA idea · Geometry, trigonometry & Voronoi diagrams

Where should my town put its next defibrillator? A Voronoi analysis

AI SLAI HL Solid Also in: Health & biology

Research question

Using a Voronoi diagram of the existing public defibrillators in [my town], where is the point furthest from all of them, and is that the best place for a new one once population and access are considered?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

Survival after cardiac arrest drops quickly with every minute before defibrillation, so distance really matters. The largest-empty-circle method gives a precise answer — and the real-world objections to it give you outstanding Reflection.

The mathematics you'll need

Perpendicular bisectors: midpoint, gradient, perpendicular gradient, equation
Voronoi cells and vertices
Vertices as circumcentres; the largest empty circle
Distance formula and scale conversions
HL: incremental algorithm when a site is added

Where the data comes from

Locate defibrillators from your town's public register or OpenStreetMap (tag emergency=defibrillator); overlay a grid with a known scale.

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 why the problem matters and state the aim.
Set up coordinates and scale; tabulate sites.
Show one perpendicular bisector in full; tabulate the rest.
Construct the diagram and find the vertex furthest from its sites.
Critique the proposed site (roads, rivers, where people are) and propose an improved one.

Pitfalls that cost marks

Repeating identical bisector calculations — show one, tabulate the rest.
A diagram with no scale or axes.
Treating straight-line distance as travel distance without comment.

Showing personal engagement

Walk the proposed route and time it.
Talk to someone who manages first-aid provision locally.
Compare the result with population density data.

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	Perpendicular bisectors: midpoint, gradient, perpendicular gradient, equation; Voronoi cells and vertices
AI HL	Good fit	Perpendicular bisectors: midpoint, gradient, perpendicular gradient, equation; Voronoi cells and vertices

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: repeating identical bisector calculations — show one, tabulate the rest; and a diagram with no scale or axes. Say how each one affects your answer.

Use of mathematics (E)

SL: perpendicular bisectors: midpoint, gradient, perpendicular gradient, equation; voronoi cells and vertices; vertices as circumcentres; the largest empty circle; distance formula and scale conversions, each explained in your own words and carried out correctly.

HL adds: incremental algorithm when a site is added, 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

Recompute with taxicab distance for a grid-like town, or weight sites by opening hours (some are inside buildings that close).

Extending it for HL

This idea already has HL mathematics in it: incremental algorithm when a site is added. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Where should our town put its next defibrillator? A Voronoi analysis (AI SL) explores a question like this one, 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?

Perpendicular bisectors: midpoint, gradient, perpendicular gradient, equation, voronoi cells and vertices, vertices as circumcentres; the largest empty circle, distance formula and scale conversions and hL: incremental algorithm when a site is added.

Where can I get data for this IA?

Locate defibrillators from your town's public register or OpenStreetMap (tag emergency=defibrillator); overlay a grid with a known scale. Checked sources and tools: OpenStreetMap and GeoGebra.

Turn this idea into your IA

The planner opens with “Where should my town put its next defibrillator? A Voronoi analysis” filled in: the question, the maths and the data sources.

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