IA idea · Trigonometry, navigation & surveying

How far can you see from the top of a hill?

AA SLAI SLAA HL Solid Also in: Geometry & Voronoi, Modelling

Research question

How far is the horizon from a given height, how well does the approximation d ≈ √(2Rh) work, and do the distances to landmarks you can actually see from a local high point agree with it?

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

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Why it makes a good exploration

A geometry derivation, an approximation with an error you can measure, and a real-world check from a viewpoint you visit.

The mathematics you'll need

Tangent to a circle and Pythagoras
Approximation using h ≪ R; error of the approximation
Arc length versus straight-line distance
Distances to landmarks from map coordinates
HL: how atmospheric refraction changes the answer (as a stated correction)

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

Visit a hill, tower or tall building; record which landmarks you can see and find their distances and heights from OpenStreetMap.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.

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

Derive the exact and approximate formulas.
Measure the approximation's error.
Visit a viewpoint and record visible landmarks.
Compare with predictions, including landmark heights.
Reflect on refraction, haze and terrain.

Pitfalls that cost marks

Forgetting that a tall landmark can be seen beyond the horizon.
Using the approximation without checking it.
Unit errors between metres and kilometres.

Showing personal engagement

Choose your own viewpoint.
Predict what you'll see before going.
Compare two heights.

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	Tangent to a circle and Pythagoras; Approximation using h ≪ R; error of the approximation
AA HL	Good fit	Tangent to a circle and Pythagoras; Approximation using h ≪ R; error of the approximation
AI SL	Good fit	Tangent to a circle and Pythagoras; Approximation using h ≪ R; error of the approximation
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: Solid. Needs some independent work beyond class examples. 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: forgetting that a tall landmark can be seen beyond the horizon — 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

Account for the height of the landmark as well as the observer, and find the furthest visible point.

Extending it for HL

This idea already has HL mathematics in it: how atmospheric refraction changes the answer (as a stated correction). 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 can we launch? Modelling the tide at our sailing club's slipway (AA 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.

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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).