Voronoi diagrams are one of the few topics that exist only in IB Maths Applications and Interpretation – students won't find them in CBSE, ICSE or IGCSE. They model a simple real question: *which point is nearest?* – the nearest hospital, school, phone tower or petrol pump. Once the geometry is clear, Voronoi questions are very scoreable.
The vocabulary
- Sites: the given points (for example, hospitals).
- Cells: the region of all points closer to one site than to any other.
- Edges: boundaries between two cells – each lies on the perpendicular bisector of the two sites.
- Vertices: where three or more edges meet – a point equidistant from three or more sites.
The key skill: perpendicular bisectors
Almost every Voronoi calculation comes back to the perpendicular bisector of two sites A and B:
- Find the midpoint of AB.
- Find the gradient of AB, then the negative reciprocal for the perpendicular.
- Write the equation of the line through the midpoint with that gradient.
Students who are fast and accurate with this pass most Voronoi questions. Students who aren't, lose marks everywhere.
Typical exam questions
- Identify the cell a point lies in, and interpret it ("which hospital is nearest?").
- Find the equation of an edge between two sites.
- Find a vertex by intersecting two edges – then check it's equidistant from three sites.
- Add a new site to a diagram: draw its perpendicular bisectors with nearby sites and remove the parts of old edges that are now closer to the new site.
- Nearest-neighbour interpolation: estimate a value at a point using the value at the site of its cell.
The "toxic waste dump" problem
A classic question: where should something unpleasant be placed to be as far as possible from the nearest site, within a region? The answer is at a vertex of the Voronoi diagram (or on the boundary of the region) – the point that maximises the minimum distance. Students should find the candidate vertices, calculate the distance to the nearest sites, and choose the largest.
Common errors
- Using the line through the two sites instead of the perpendicular bisector.
- Arithmetic slips with negative reciprocals.
- Forgetting to erase old edge segments when adding a site.
- Not checking that a vertex is equidistant from all three sites.
- Giving a coordinate without interpreting it in context.
How to practise
- Perpendicular bisector drills until they take under two minutes.
- Draw diagrams by hand on squared paper for three or four sites.
- Use graphing software or a GDC to check edges, but always show the working.
- Past-paper questions, especially "add a site" and toxic waste dump questions.
How Maths Bodhi helps AI students
- A quick check of coordinate geometry (midpoints, gradients, line equations) – the real foundation.
- Step-by-step diagram building, with the student drawing and explaining.
- Context interpretation practice for the final parts of questions.
- Past IB questions marked against the markscheme.
Ajay Vatsyayan (M.Tech, 14+ years teaching maths) teaches IB DP maths at Maths Bodhi and has taught students from schools across Gurgaon, including DPS International, GD Goenka World School, Suncity School and Pathways World School. Maths Bodhi is independent and not affiliated with any school. Read Ajay's profile, or see our IB geometry tutor, AI SL and AI HL pages.
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Related: AI SL modelling with functions, AI SL statistics and the IB DP maths guide for Gurgaon families.
Course content checked against the current IB AI guide by the Maths Bodhi team (October 2026). Reviewed by Ajay Vatsyayan.
Frequently asked questions
- Are Voronoi diagrams in IB maths AA?
- No. Voronoi diagrams are part of the IB Maths Applications and Interpretation (AI) course.
- What is an edge in a Voronoi diagram?
- An edge is a boundary between two cells, and it lies on the perpendicular bisector of the two sites.
- What is the toxic waste dump problem?
- Finding the point in a region that is as far as possible from the nearest site. The answer lies at a vertex of the Voronoi diagram or on the region's boundary.