How big is my local lake? Estimating area with the trapezoidal rule

AI SLAI HLAA SLAA HL Accessible Also in: Geometry & Voronoi, Environment

Research question

How does the estimate of [a lake]'s area from the trapezoidal rule change as the number of strips increases, and how close does it get to the officially published area?

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

Why it makes a good exploration

Irregular shapes have no area formula. The trapezoidal rule, applied to a real lake, island or park measured from a map, shows why numerical methods exist and how accuracy improves with more strips.

The mathematics you'll need

The trapezoidal rule
Coordinate geometry from a scaled map
Error as a function of strip width
HL: fitting functions to the boundary and integrating

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

Use OpenStreetMap or a satellite map with a scale bar; overlay a grid and read boundary widths.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

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

Choose a lake with a published area.
Set up axes and read widths at equal intervals.
Apply the trapezoidal rule for 5, 10, 20 and 40 strips.
Plot the estimate against strip number and analyse convergence.
Discuss map scale, projection and seasonal water levels.

Pitfalls that cost marks

Inconsistent scale conversion.
Too few strips to see a trend.
Not explaining why the rule over- or under-estimates on curved edges.

Showing personal engagement

Choose a lake or park that matters to you.
Compare with a pixel-counting estimate.
Predict the error before calculating it.

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

Taking it further

Compare with Simpson's rule (new maths — explain it) and discuss which converges faster and why.

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).