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

Can you predict a tree's height from its trunk?

AI SLAI HLAA SL Solid Also in: Geometry & Voronoi, Environment, Statistics

Research question

Is the height of trees in my local park related to trunk diameter by a power law, and does the exponent agree with the value predicted by the theory of elastic similarity (about ⅔)?

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Why it makes a good exploration

Foresters estimate height from diameter all the time, and biologists argue about the exponent. You measure heights with trigonometry and a clinometer, then test a published scientific claim with your own data.

The mathematics you'll need

Trigonometry (right-angled triangles) to measure heights
Power models and log-log regression
Correlation and residuals
Measurement uncertainty

Modelling the data, step by step

This idea usually needs a power model. See it worked step by step, with a criterion tip at every step: Power y = a·xⁿ (log–log) · Choosing and comparing models.

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 30+ trees of one species: circumference at 1.3 m and height using a clinometer app and pacing the distance.

phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.

A possible outline

Explain the clinometer method and its error.
Collect and tabulate data for one species.
Fit a power model via logs; compare the exponent with ⅔.
Examine outliers (damaged or crowded trees).
Reflect on species, age and environment.

Pitfalls that cost marks

Mixing species, which have different shapes.
Not adding eye height to the trigonometric height.
Measuring height from too close, where small angle errors explode.

Showing personal engagement

Choose trees you know — your school grounds or a local park.
Calculate how your height error depends on your distance from the tree and choose the best distance.
Compare two species.

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	Trigonometry (right-angled triangles) to measure heights; Power models and log-log regression
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	Trigonometry (right-angled triangles) to measure heights; Power models and log-log regression
AI HL	Good fit	Trigonometry (right-angled triangles) to measure heights; Power models and log-log regression

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

Reflection (D)

For this idea, start with the weak points: mixing species, which have different shapes; and not adding eye height to the trigonometric height. Say how each one affects your answer.

Use of mathematics (E)

SL: trigonometry (right-angled triangles) to measure heights; power models and log-log regression; correlation and residuals; measurement uncertainty, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Use your model to estimate the height of a tree you cannot measure and assess the prediction interval.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL modelling exploration adds.

See a complete IA, marked

When can we launch? Modelling the tide at our sailing club's slipway (AA SL) is a complete modelling exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI SL, AI HL and AA SL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Trigonometry (right-angled triangles) to measure heights, power models and log-log regression, correlation and residuals and measurement uncertainty.

Where can I get data for this IA?

Measure 30+ trees of one species: circumference at 1.3 m and height using a clinometer app and pacing the distance. Checked sources and tools: phyphox (RWTH Aachen).

Turn this idea into your IA

The planner opens with “Can you predict a tree's height from its trunk?” filled in: the question, the maths and the data sources.

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