Voronoi diagrams are one of the more visually distinctive topics in the IB Math AI syllabus — and one that a lot of students initially find confusing simply because it looks so different from anything else in the course. Once the core idea clicks, though, it's one of the more mechanical, learnable topics: the questions follow a fairly predictable structure once you know the vocabulary.
The core idea
A Voronoi diagram divides a plane into regions based on distance to a set of points, called sites. Each region (called a cell) contains every point on the plane that is closer to its own site than to any other site. In practice, this comes up in questions about things like which fire station would respond fastest to a given location, or which shop is nearest to a customer — the "closest point" framing is what makes this topic feel applied rather than purely abstract.
Key vocabulary you need cold
- Site — one of the original points the diagram is built around (e.g. a shop, a hospital)
- Cell — the region of the plane closest to a particular site
- Edge — the boundary between two neighbouring cells; every edge lies on the perpendicular bisector of the two sites it separates
- Vertex — a point where three or more edges meet; a vertex is equidistant from all the sites whose cells meet there
Almost every calculation in this topic comes back to finding the perpendicular bisector between two points: find the midpoint, find the gradient of the segment joining the two sites, then use the negative reciprocal gradient through that midpoint. If you're confident with perpendicular bisectors from coordinate geometry, most of this topic is just applying that one skill repeatedly.
Typical question structure
- Construct part of a diagram. You're usually given coordinates for 3-5 sites and asked to find the equations of one or more edges (perpendicular bisectors) between pairs of sites.
- Find a vertex. Once you have two edges, their intersection point is a vertex — solve the two bisector equations simultaneously.
- Interpret a cell in context. A common final part asks something like "which site is a given point closest to" — this is where the real-world framing (nearest hospital, nearest store) comes in, and it's usually the most straightforward part if the earlier construction was done correctly.
- The "toxic waste dump" / largest empty circle problem. A distinctive AI HL extension: finding the point that is as far as possible from all sites, which occurs at a vertex of the diagram (or on the boundary) — a favourite for a final, harder mark.
Where marks are commonly lost
The most frequent error isn't conceptual — it's an arithmetic slip finding the midpoint or the perpendicular gradient. Because every part of the question tends to build on the previous edge equation, one early arithmetic error can cascade through the rest of the question. Double-checking your perpendicular bisector equation by confirming it's equidistant from both original sites (substitute the midpoint back in) is a fast way to catch this before it costs multiple marks.
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✦ Try it freeFrequently asked questions
Is Voronoi diagrams tested at AI SL too?
The core Voronoi diagram concept is an AI HL topic; SL students are not typically assessed on it — always confirm against your current subject guide, as syllabus scope can be updated.
Do I need a calculator for Voronoi diagram questions?
Yes — like the rest of AI, a GDC is expected and used throughout, particularly for solving simultaneous equations when finding vertices.