Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

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, Trig & navigation

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?

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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

Modelling the data, step by step

This idea usually needs a sinusoidal model. See it worked step by step, with a criterion tip at every step: Sinusoidal y = a sin(b(x − c)) + d.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for 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.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Right-angled trigonometry: elevation = arctan(h/s); Modelling elevation against time with a sinusoid
AA HL	Fits, but add an HL technique	Right-angled trigonometry: elevation = arctan(h/s); Modelling elevation against time with a sinusoid
AI SL	Good fit	Right-angled trigonometry: elevation = arctan(h/s); Modelling elevation against time with a sinusoid
AI HL	Fits, but add an HL technique	Right-angled trigonometry: elevation = arctan(h/s); Modelling elevation against time with a sinusoid

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

Reflection (D)

For this idea, start with the weak points: stick not vertical; and too short a time window to see the maximum. Say how each one affects your answer.

Use of mathematics (E)

SL: 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, each explained in your own words and carried out correctly.

How geometry & voronoi explorations reach the top bands in general: Geometry, trigonometry & Voronoi diagrams IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

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

Extending it for HL

Add an HL technique the question genuinely needs: what an HL geometry & voronoi exploration adds.

See a complete IA, marked

When is the sea warm enough for our swimming club? A sinusoidal model of sea temperature (AI SL) is a complete geometry & voronoi exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AI SL, AA HL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Right-angled trigonometry: elevation = arctan(h/s), modelling elevation against time with a sinusoid, finding maxima (calculus or technology) and latitude from noon elevation and the date.

Where can I get data for this IA?

Measure a vertical stick's shadow every 20 minutes from mid-morning to mid-afternoon; check with sun-position data. Checked sources and tools: timeanddate.com sun calculator and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Modelling the sun's height from a stick's shadow” filled in: the question, the maths and the data sources.

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