Voronoi diagrams — construction and applications (IB Math AI HL)
Updated October 2026
Quick answer: Each cell of a Voronoi diagram holds the points closer to its site than to any other site. The edges lie on perpendicular bisectors of pairs of sites, and each vertex is the same distance from three or more sites.
Voronoi diagrams are SL content (AI SL 3.6) that HL AI students sit too: build cells from perpendicular bisectors, add a new site to an existing diagram, and solve the 'toxic waste dump' problem — the point furthest from every site.
Practise Voronoi diagrams — construction and applications with AI marking
Open the full HL AI engine — every question in this unit is worked through with instant AI-graded feedback on your handwritten workings.
A cell contains every point nearer to its site than to any other site.
Edges lie on perpendicular bisectors: through the midpoint of two sites, with gradient −1 ÷ (gradient of the line joining them).
A vertex is equidistant from (at least) three sites, so it is where their bisectors meet.
Adding a site: draw the bisectors between the new site and its neighbours, then remove the parts of the old edges that are now inside the new cell.
The toxic waste dump problem: the point furthest from every site, inside a boundary, is at a vertex of the diagram or on the boundary. Compare the distances at each candidate.
Worked example
Three sites are A(0, 0), B(6, 0) and C(0, 4). Find the equations of the perpendicular bisectors of AB, AC and BC, the vertex of the Voronoi diagram, and its distance from each site.
AB: midpoint (3, 0), AB is horizontal, so the bisector is x = 3.
AC: midpoint (0, 2), AC is vertical, so the bisector is y = 2.
BC: midpoint (3, 2), gradient of BC = −4/6 = −2/3, so the bisector has gradient 3/2: y − 2 = (3/2)(x − 3).
The bisectors meet at (3, 2), which is on all three lines.
Distance from (3, 2) to A = √(9 + 4) = √13 ≈ 3.61, and the same to B and to C.
Answer: x = 3, y = 2 and y = 1.5x − 2.5; vertex (3, 2), √13 ≈ 3.61 from each site.
Find the perpendicular bisectors of two pairs of the three sites and solve them simultaneously. The point is the same distance from all three sites, so check with the third bisector.
Where can I practise voronoi diagrams questions?
On the Scalar Products and Voronoi questions page: 32 exam-style HL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for voronoi diagrams?
Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working.