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

IA idea · Geometry, trigonometry & Voronoi diagrams

Why do flight paths curve on the map?

AA SLAA HLAI SLAI HL Ambitious Also in: Networks & graphs, Pure maths, Trig & navigation

Research question

How much shorter is the great-circle route between [two cities] than the route that appears straight on a Mercator map, and how does the saving depend on latitude?

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Why it makes a good exploration

Long-haul flights seem to curve towards the poles on maps. Showing, with vectors or spherical trigonometry, that these routes are actually the shortest is a beautiful exploration of 3-D geometry.

The mathematics you'll need

Latitude/longitude to 3-D coordinates
Vectors and the angle between them (HL) or the spherical law of cosines
Arc length
Comparing with a rhumb line (constant bearing)

Where the data comes from

City coordinates from OpenFlights; optionally real flight distances from airline schedules.

OpenFlights data — Airports (with latitude/longitude) and airline routes as CSV.

Real data for this idea

Primary source: OpenFlights data: Airports and routes. airports.dat gives name, IATA code, latitude and longitude; routes.dat gives each route's source and destination airport. Licence: Open Database Licence. The routes file is a 2014 snapshot.

The OpenFlights routes file is a 2014 snapshot: treat it as historical and say so. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Convert coordinates to 3-D vectors.
Find the angle between vectors and the great-circle distance.
Compute the rhumb-line distance for comparison.
Repeat for routes at different latitudes.
Reflect on winds, airspace and the flat-map illusion.

Pitfalls that cost marks

Mixing degrees and radians.
Using the haversine formula as a black box.
Comparing with a ruler distance on a Mercator map without explaining the projection.

Showing personal engagement

Use a flight you have taken.
Plot both routes on a globe with string.
Explain why Europe–North America flights pass near Greenland.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Latitude/longitude to 3-D coordinates; Vectors and the angle between them (HL) or the spherical law of cosines
AA HL	Good fit	Latitude/longitude to 3-D coordinates; Vectors and the angle between them (HL) or the spherical law of cosines
AI SL	Fits — ambitious at SL	Latitude/longitude to 3-D coordinates; Vectors and the angle between them (HL) or the spherical law of cosines
AI HL	Good fit	Latitude/longitude to 3-D coordinates; Vectors and the angle between them (HL) or the spherical law of cosines

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: mixing degrees and radians; and using the haversine formula as a black box. Say how each one affects your answer.

Use of mathematics (E)

SL: latitude/longitude to 3-D coordinates; vectors and the angle between them (HL) or the spherical law of cosines; arc length; comparing with a rhumb line (constant bearing), 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

Derive the maximum latitude reached by a great-circle route between two cities.

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 AA SL, AA HL, AI SL and AI 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?

Latitude/longitude to 3-D coordinates, vectors and the angle between them (HL) or the spherical law of cosines, arc length and comparing with a rhumb line (constant bearing).

Where can I get data for this IA?

City coordinates from OpenFlights; optionally real flight distances from airline schedules. Checked sources and tools: OpenFlights data.

Turn this idea into your IA

The planner opens with “Why do flight paths curve on the map?” filled in: the question, the maths and the data sources.

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