IA idea · Trigonometry, navigation & surveying

Measuring the Earth with a stick and a partner school

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

Research question

If you and a partner school far to the north or south measure a stick's shadow at local solar noon on the same day, what radius of the Earth do you get, and what limits its accuracy?

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 collaboration with a real measurement and a known answer to compare with. The error analysis (shadow length, timing, distance between schools) is genuine and quantitative.

The mathematics you'll need

Shadow angles with tan⁻¹
Arc length and sectors in radians
Distance between latitudes
Error propagation through the formula
Solar noon from sunrise and sunset times

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 shadows yourself; partner with a school at a different latitude (your teacher may help) and find solar noon with timeanddate.com.

timeanddate.com sun calculator — Sunrise, sunset and day length for any city and date.
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

Explain the method and the geometry.
Measure shadows on the same day.
Calculate the radius.
Analyse the main sources of error.
Compare with the accepted value and reflect.

Pitfalls that cost marks

Not measuring at local solar noon.
Schools at different longitudes without correction.
A stick that isn't vertical.

Showing personal engagement

Organise the partner measurement yourself.
Repeat on several days.
Compare with Eratosthenes' method and assumptions.

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	Shadow angles with tan⁻¹; Arc length and sectors in radians
AA HL	Good fit	Shadow angles with tan⁻¹; Arc length and sectors in radians
AI SL	Good fit	Shadow angles with tan⁻¹; Arc length and sectors in radians
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: not measuring at local solar noon — 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

Use measurements at several times of day to estimate the sun's elevation curve and improve the noon value.

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 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.

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).