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

IA idea · Geometry, trigonometry & Voronoi diagrams

Should bigger hospitals get bigger regions? Weighted Voronoi diagrams

AI HLAI SL Ambitious Also in: Health & biology, Networks & graphs

Research question

If each hospital's region is weighted by its number of beds, how do the regions change compared with a standard Voronoi diagram, and does the weighted version balance patients per bed better?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

A standard Voronoi diagram treats a small clinic and a major hospital the same. Weighting sites by capacity is a real planning idea with new geometry (curved boundaries) — ideal for an ambitious AI student.

The mathematics you'll need

Standard Voronoi diagrams
Multiplicatively weighted distance and Apollonius circles (new — explain)
Estimating population per region
Comparing load per bed

Where the data comes from

Hospital locations from OpenStreetMap and bed numbers from official health-service publications; population from census data.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
Office for National Statistics (UK) — UK population, earnings, inflation, births and deaths time series.
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.
Also useful: Office for National Statistics (UK): Population estimates. Use the UK mid-year estimates time series (year and total population). Licence: Open Government Licence v3.0: free to reuse with an acknowledgement.

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 the weighted distance d/w and why it changes boundaries.
Derive the boundary between two weighted sites (a circle).
Construct both diagrams for 4–6 hospitals.
Estimate population per bed for each.
Reflect on specialities, emergency departments and travel time.

Pitfalls that cost marks

Using weights without justification.
Claiming boundaries are straight in the weighted case.
Too many sites to construct accurately.

Showing personal engagement

Use hospitals in your own region.
Discuss with a health professional how catchments are actually decided.
Compare additive and multiplicative weighting.

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	Fits — ambitious at SL	Standard Voronoi diagrams; Multiplicatively weighted distance and Apollonius circles (new — explain)
AI HL	Good fit	Standard Voronoi diagrams; Multiplicatively weighted distance and Apollonius circles (new — explain)

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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 weights without justification; and claiming boundaries are straight in the weighted case. Say how each one affects your answer.

Use of mathematics (E)

SL: standard Voronoi diagrams; multiplicatively weighted distance and Apollonius circles (new — explain); estimating population per region; comparing load per bed, 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

Test a power diagram (additively weighted) and compare with the multiplicative version.

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 HL and AI SL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Standard Voronoi diagrams, multiplicatively weighted distance and Apollonius circles (new — explain), estimating population per region and comparing load per bed.

Where can I get data for this IA?

Hospital locations from OpenStreetMap and bed numbers from official health-service publications; population from census data. Checked sources and tools: OpenStreetMap, Office for National Statistics (UK) and GeoGebra.

Turn this idea into your IA

The planner opens with “Should bigger hospitals get bigger regions? Weighted Voronoi diagrams” filled in: the question, the maths and the data sources.

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