IB Maths Voronoi Diagrams (AI): Syllabus, Exam Questions and How to Revise Them

A Voronoi diagram divides a plane into regions called cells, based on distance to a fixed set of points called sites. Every point inside a cell is closer to its own site than to any other. In IB Maths AI you construct these using perpendicular bisectors between neighbouring sites — it's Topic 3.9 in the syllabus.

Voronoi diagrams sit only in Mathematics: Applications and Interpretation — they don't appear anywhere in the Analysis and Approaches syllabus. It's one of the cleanest content differences between the two courses, alongside AI's heavier statistics and modelling load. If geometric, GDC-based reasoning appeals to your child, that's a genuine data point for choosing AI over AA.

Voronoi diagrams are SL content, so both AI SL and AI HL students study identical material — sites, cells, adding a new site, nearest-neighbour interpolation and the toxic waste dump problem. Unlike vectors or complex numbers, HL gets no extra depth here, which means SL and HL exam questions on this topic look almost interchangeable.

You need six terms cold: site, vertex, edge, cell, plus the two applications — nearest-neighbour interpolation and the toxic waste dump problem. Examiners lose patience fast with answers that mix these up, especially students who call an edge a 'boundary line' or confuse a vertex with a site.

Voronoi diagrams appear as extended-response questions in Paper 1 or Paper 2 of AI SL and HL — both papers permit a GDC, so you're expected to construct diagrams, add sites and interpolate values through reasoning, not just recall. Expect it worth roughly 6–10 marks inside a longer, multi-part question.

Most Voronoi questions sit inside one multi-part question worth 6 to 10 marks total, rarely a whole paper section on their own. Marks split across accurate construction, correctly identifying the toxic waste dump point, and applying nearest-neighbour interpolation — so a shaky bisector early on costs you marks further down the question too.

The top error is drawing a bisector that isn't genuinely perpendicular — examiners check the gradient, not a roughly-central line. Second is forgetting to erase the old edge once a new site is added. Third is mixing up the toxic waste dump point with nearest-neighbour interpolation — they use opposite logic.

Plot every site, then for each pair of neighbouring sites draw the perpendicular bisector of the segment joining them. Keep only the portion of each bisector closest to both relevant sites — where three bisectors meet, you've found a vertex. Repeat until every site sits inside its own bounded or partly-bounded cell.

It asks you to find the point inside a Voronoi diagram that is as far as possible from its nearest site — the safest spot to place something hazardous. That point is always a vertex of the diagram (occasionally a boundary point), found by comparing the distance from each candidate vertex to its nearest site.

Plot the new site, then draw perpendicular bisectors between it and every existing site whose cell might now be split. Erase the parts of the old edges that fall on the new site's side of each new bisector, and trim the new bisectors where they cross other cell boundaries — the diagram must still tile the plane exactly.

Nearest-neighbour interpolation estimates an unknown value at a location by assigning it the value already known at the closest site — whichever Voronoi cell contains that location tells you which site to use. It's a common IB context for estimating rainfall, temperature or population density between sparsely spaced measuring stations.

No — Voronoi diagrams are visual and GDC-friendly, but that doesn't make the whole AI course softer than AA. AI trades AA's algebraic proof and calculus depth for heavier statistics, modelling and technology use. A student who finds geometric reasoning intuitive may find this one topic straightforward without the overall course being any easier.

The fastest gains come from repeated construction practice, not re-reading notes — have your child build diagrams from scratch with a ruler and GDC, then time a full multi-part past-paper question. According to the IB, first exams for the current Applications and Interpretation guide were held in 2021, and Voronoi content has featured in some form most sessions since.

Prioritise resources that pair worked constructions with genuine exam-style multi-part questions, not just definitions and vocabulary lists. Practising the full sequence — construct, add a site, locate the toxic waste dump point, interpolate a value — matters far more than memorising terms. Topical worksheets and mark schemes focused specifically on AI Topic 3 are the most efficient use of limited revision time.