IA idea · Trigonometry, navigation & surveying

How far off do you end up? Navigating a route by bearings

AA SLAI SL Accessible Also in: Geometry & Voronoi

Research question

If you walk a closed route by compass bearings and paced distances, how far from the start do you finish, how does that error depend on the number of legs, and which kind of error (angle or distance) contributes most?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A practical navigation skill with a measurable closing error. Breaking the error down into its sources is a satisfying analysis with vectors or trigonometry.

The mathematics you'll need

Bearings and the cosine and sine rules
Resolving legs into east and north components
Closing error as a distance and bearing
Simulating random errors
Comparing angle and distance errors

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Walk a route on a playing field with a compass app and pacing; record each leg; repeat several times.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Plan a closed route and calculate it should return to the start.
Walk it several times and record the finishing point.
Calculate the closing error each time.
Model errors in angle and distance and compare with observations.
Reflect on which error to reduce first.

Pitfalls that cost marks

Calibrating pace length only once.
Magnetic interference near buildings.
Treating one walk as the answer.

Showing personal engagement

Walk it yourself and with friends.
Predict the error before walking.
Design the route that minimises error.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Bearings and the cosine and sine rules; Resolving legs into east and north components
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	Bearings and the cosine and sine rules; Resolving legs into east and north components
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Take the measurements yourself, choose the place, and compare at least two methods. Explaining why you chose each position or angle is engagement an examiner can see.

Reflection (D)

Analyse measurement error: how much does a small error in an angle change the answer, which method is most robust, and would you trust the result for the decision you are making? For this idea, start with: calibrating pace length only once — say how it affects your answer.

Use of mathematics (E)

SL: Right-angled trigonometry, sine and cosine rules, bearings and 3D trigonometry applied correctly to your own measurements, with an error analysis.

HL: Error propagation with derivatives, vectors in three dimensions, spherical geometry derived rather than quoted, or an optimisation of where to measure from.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Simulate thousands of walks with random errors and compare the distribution with your real results.

Extending it for HL

Use derivatives to find how sensitive the result is to each measured angle, and choose the measuring position that minimises the error.

See a complete IA, marked

Our annotated exemplar When is the sea warm enough for our swimming club? A sinusoidal model of sea temperature (AI SL) asks a different question, but shows how a complete trig & navigation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

Get the free checklist

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).