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

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

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

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?

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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

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.

Real data for this idea

Primary source: OpenStreetMap. Read off coordinates (right-click, “Show address”) or use Export for a small area. Licence: Open Database Licence: credit “© OpenStreetMap contributors”; data you build from it must be shared alike.

OpenStreetMap is under the Open Database Licence: credit “© OpenStreetMap contributors”. Links checked October 2026; we link to the data and never copy it. More free data sets.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	The trapezoidal rule; Coordinate geometry from a scaled map
AA HL	Fits, but add an HL technique	The trapezoidal rule; Coordinate geometry from a scaled map
AI SL	Good fit	The trapezoidal rule; Coordinate geometry from a scaled map
AI HL	Fits, but add an HL technique	The trapezoidal rule; Coordinate geometry from a scaled map

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: inconsistent scale conversion; and too few strips to see a trend. Say how each one affects your answer.

Use of mathematics (E)

SL: the trapezoidal rule; coordinate geometry from a scaled map; error as a function of strip width, each explained in your own words and carried out correctly.

HL adds: fitting functions to the boundary and integrating, each explained in your own words and carried out correctly.

How calculus explorations reach the top bands in general: Calculus & optimisation IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

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

Extending it for HL

This idea already has HL mathematics in it: fitting functions to the boundary and integrating. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) is a complete calculus exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

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

What mathematics does this IA use?

The trapezoidal rule, coordinate geometry from a scaled map, error as a function of strip width and hL: fitting functions to the boundary and integrating.

Where can I get data for this IA?

Use OpenStreetMap or a satellite map with a scale bar; overlay a grid and read boundary widths. Checked sources and tools: OpenStreetMap and GeoGebra.

Turn this idea into your IA

The planner opens with “How big is my local lake? Estimating area with the trapezoidal rule” filled in: the question, the maths and the data sources.

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