IA idea · Trigonometry, navigation & surveying

How wide is the river? Measuring a distance you can't cross

AA SLAI SL Accessible Also in: Geometry & Voronoi

Research question

Which of three trigonometric methods for measuring the width of a river (or a road, or a sports field) from one side gives the most reliable answer, and how sensitive is each to a one-degree error in an angle?

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

The measurement is real, the methods are in the SL course, and the error analysis gives every method a score you can compare. Checking against a map or a safe direct measurement gives a true value.

The mathematics you'll need

Right-angled trigonometry
Sine rule with a measured baseline
Error analysis: recalculating with ±1° and ±0.1 m
HL: derivatives to approximate error
Comparing methods with a chosen measure of reliability

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

Measure angles with a clinometer or phone app and a tape for the baseline; check against OpenStreetMap or a measurement where it is safe.

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.

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

Explain three methods and draw each.
Take repeated measurements for each.
Calculate widths and their spread.
Analyse sensitivity to angle and baseline errors.
Decide which method you trust and why.

Pitfalls that cost marks

One measurement per method.
No independent check of the true width.
Unsafe measuring near water; stay well back from the edge.

Showing personal engagement

Use a place you know.
Choose the baseline length that minimises error and explain why.
Compare your answer with a map.

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	Right-angled trigonometry; Sine rule with a measured baseline
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	Right-angled trigonometry; Sine rule with a measured baseline
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: one measurement per method — 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

Find the optimal baseline length analytically with calculus.

Extending it for HL

This idea already has HL mathematics in it: derivatives to approximate error. 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.

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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).