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

IA idea · Geometry, trigonometry & Voronoi diagrams

Voronoi diagrams in a grid city: Euclidean or taxicab distance?

AI SLAI HLAA HL Ambitious Also in: Pure maths, Networks & graphs

Research question

How different are the Voronoi regions of the fire stations in [a grid-plan city] under Euclidean and taxicab (Manhattan) distance, and which better predicts the real nearest station by road?

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Why it makes a good exploration

In a city with a street grid, vehicles cannot drive diagonally. Taxicab distance changes the shape of Voronoi cells completely — and comparing both with real road distances is a genuinely original analysis.

The mathematics you'll need

Euclidean and taxicab metrics
Bisectors under the taxicab metric (piecewise lines)
Voronoi construction by hand and technology
Testing predictions with sampled points

Where the data comes from

Station coordinates from OpenStreetMap for a city with a grid layout; check nearest stations by road using an online map for 20 sample points.

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

Introduce the taxicab metric and its “circles”.
Derive the shape of a taxicab bisector.
Build both diagrams.
Test 20 sample addresses against real road distances.
Reflect on one-way streets, rivers and diagonal avenues.

Pitfalls that cost marks

Choosing a city without a grid.
Drawing taxicab bisectors as straight lines.
Sampling points in a biased way.

Showing personal engagement

Choose a city you have visited.
Predict which metric will win and why.
Draw a taxicab circle and explain what it means for response time.

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	Good fit	Euclidean and taxicab metrics; Bisectors under the taxicab metric (piecewise lines)
AI SL	Fits — ambitious at SL	Euclidean and taxicab metrics; Bisectors under the taxicab metric (piecewise lines)
AI HL	Good fit	Euclidean and taxicab metrics; Bisectors under the taxicab metric (piecewise lines)

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: choosing a city without a grid; and drawing taxicab bisectors as straight lines. Say how each one affects your answer.

Use of mathematics (E)

SL: euclidean and taxicab metrics; bisectors under the taxicab metric (piecewise lines); voronoi construction by hand and technology; testing predictions with sampled points, 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

Consider a rotated grid or a city with major diagonals and define a hybrid metric.

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

What mathematics does this IA use?

Euclidean and taxicab metrics, bisectors under the taxicab metric (piecewise lines), voronoi construction by hand and technology and testing predictions with sampled points.

Where can I get data for this IA?

Station coordinates from OpenStreetMap for a city with a grid layout; check nearest stations by road using an online map for 20 sample points. Checked sources and tools: OpenStreetMap and GeoGebra.

Turn this idea into your IA

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