IA idea · Simulation & Monte Carlo methods

Estimating the area of a leaf with random points

AA SLAI SLAA HLAI HL Accessible Also in: Calculus, Health & biology

Research question

How accurately can random points estimate the area of a leaf (or any irregular shape), how does the error shrink as the number of points grows, and how does Monte Carlo compare with counting squares and the trapezoidal rule?

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

It is a simple, visual introduction to Monte Carlo methods with a real object, and the error question (why does it shrink like 1/√n?) leads to genuinely interesting mathematics.

The mathematics you'll need

Probability as a proportion of area
Binomial model for the number of hits
Standard error ∝ 1/√n, checked against repeated estimates
The trapezoidal rule from measured widths
Comparing methods for accuracy and effort

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.

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Binomial and Poisson models. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Photograph or scan leaves on squared paper; place random points with a spreadsheet and count hits by eye or in GeoGebra.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
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

Measure the leaf area by counting squares as a reference.
Estimate it with random points for several sample sizes.
Repeat each size many times and measure the spread.
Compare the spread with the binomial prediction.
Compare with the trapezoidal rule and reflect on which you would use.

Pitfalls that cost marks

Using one estimate per sample size, so the error pattern is invisible.
Forgetting the area of the bounding rectangle in the scale factor.
No reference value to judge accuracy against.

Showing personal engagement

Use leaves from a plant you grow or a tree you pass.
Predict how many points you need for 1% accuracy, then test it.
Compare a smooth leaf with a jagged one.

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	Probability as a proportion of area; Binomial model for the number of hits
AA HL	Fits, but add an HL technique	Probability as a proportion of area; Binomial model for the number of hits
AI SL	Good fit	Probability as a proportion of area; Binomial model for the number of hits
AI HL	Fits, but add an HL technique	Probability as a proportion of area; Binomial model for the number of hits

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

Personal engagement (C)

Design the simulation yourself: choose the rules, test them on a small case you can check by hand, and change one rule at a time to answer a question you care about.

Reflection (D)

Compare simulation with exact theory or real data, say how many runs you used and how much the answer varies between batches, and question the random-number assumptions. For this idea, start with: using one estimate per sample size, so the error pattern is invisible — say how it affects your answer.

Use of mathematics (E)

SL: A probability model described precisely, simulated correctly, with the simulated answer compared with an exact calculation for at least one simple case and the results summarised with appropriate statistics.

HL: An estimate of the simulation's error (standard error, or a confidence interval for the estimate), a distribution fitted to the results and tested, or an exact result proved for the general case that the simulation confirms.

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 the same idea to estimate π or a volume, or compare random points with a regular grid of points and explain the difference.

Extending it for HL

Give a confidence interval for each simulated estimate and show how it narrows as the number of runs grows, or prove a general result that the simulation confirms.

See a complete IA, marked

Our annotated exemplar Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) asks a different question, but shows how a complete simulation 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

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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).