IA idea · Trigonometry, navigation & surveying

Measuring the area of an irregular field with triangles

AA SLAI SL Accessible Also in: Geometry & Voronoi, Calculus

Research question

How accurately can the area of an irregular plot be found by splitting it into triangles with measured sides and angles, compared with the trapezoidal rule from offsets and with the area from a map?

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

Real surveyors did this for centuries. Comparing three methods with a map value gives a clear verdict and a natural error analysis.

The mathematics you'll need

Area = ½ab sin C and Heron's formula
Cosine rule to check consistency
The trapezoidal rule from offsets
Shoelace formula from map coordinates (new)
Percentage errors and their sources

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 a school field, garden or park plot with a tape and a phone clinometer/compass; read corner coordinates from OpenStreetMap.

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

Split the plot into triangles and measure.
Calculate the area two ways.
Use offsets and the trapezoidal rule.
Compare with the map area.
Reflect on which measurements carry most error.

Pitfalls that cost marks

Triangles that are very thin (large angle errors).
Not checking the map's accuracy.
No error estimate.

Showing personal engagement

Survey a place you use.
Choose the triangulation that minimises error.
Compare with the owner's stated area.

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	Area = ½ab sin C and Heron's formula; Cosine rule to check consistency
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	Area = ½ab sin C and Heron's formula; Cosine rule to check consistency
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: triangles that are very thin (large angle errors) — 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

Propagate measurement errors through each formula to find the most robust method.

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.

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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).