IA idea · Geometry, trigonometry & Voronoi diagrams

Modelling the sun's height from a stick's shadow

AA SLAI SLAA HLAI HL Accessible Also in: Modelling, Environment

Research question

Can the length of a vertical stick's shadow over a day be modelled with trigonometric functions to estimate solar noon, the sun's maximum elevation and my latitude?

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

Why it makes a good exploration

Eratosthenes measured the Earth with shadows. You can use a stick in the garden to find your latitude and local solar noon — a satisfying mix of trigonometry, modelling and astronomy.

The mathematics you'll need

Right-angled trigonometry: elevation = arctan(h/s)
Modelling elevation against time with a sinusoid
Finding maxima (calculus or technology)
Latitude from noon elevation and the date

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 vertical stick's shadow every 20 minutes from mid-morning to mid-afternoon; check with sun-position data.

timeanddate.com sun calculator — Sunrise, sunset and day length for any city and date.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

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 geometry of shadows.
Collect data on a clear day.
Fit a model to elevation over time.
Estimate solar noon and latitude; compare with the true values.
Reflect on uneven ground, stick tilt and a sinusoid being an approximation.

Pitfalls that cost marks

Stick not vertical.
Too short a time window to see the maximum.
Ignoring the sun's declination on the date.

Showing personal engagement

Compare with a friend's measurements in another city.
Repeat on another date and compare.
Reproduce Eratosthenes' calculation with the two sets of data.

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

Taking it further

Use two locations' simultaneous measurements to estimate the Earth's circumference.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).