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IB Maths AI SL · Unit 3: Geometry and Trigonometry

Voronoi Diagrams (IB Maths AI SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

A Voronoi diagram splits a region into cells, one around each site, so that every point in a cell is closer to that site than to any other. The edges lie on perpendicular bisectors of pairs of sites and the vertices are equidistant from three sites. In AI SL you find bisector equations, add a site to a diagram, and solve the "toxic waste dump" problem: the point furthest from every site.

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Start with one question on voronoi diagrams. Stuck? Take a hint. No sign-in needed.

Sites A(1, 1), B(7, 3) and C(3, 7). Find the Voronoi vertex where the three cells meet.
Choose your answer · keys 1–4

Voronoi Diagrams notes

Sites, cells, edges and vertices

Each site owns a cell: the points nearer to it than to any other site. An edge separates two cells and is part of the perpendicular bisector of their two sites. A vertex is where three cells meet, so it is the same distance from three sites.

Equation of a perpendicular bisector

Find the midpoint of the two sites, the gradient between them, then use the negative reciprocal. For sites A(2,1) and B(6,3): midpoint (4,2), gradient 12, so the bisector has gradient −2 and equation y=−2x+10.

Nearest neighbour interpolation

To estimate a value at a point, use the value of the site whose cell the point lies in. This is how a Voronoi diagram answers "which hospital is closest?" or "estimate the rainfall here".

Adding a site

Draw the perpendicular bisectors between the new site and its neighbours, and keep only the parts that form the new cell's boundary. The cells next to the new site shrink; cells far away do not change.

The toxic waste dump problem

The point inside a region that is as far as possible from every site lies at a vertex of the diagram or on the boundary of the region. Check the candidate points and work out the distance to the nearest site for each; the largest of these wins.

Key formulas

Distance; midpoint In the booklet
d=(x1−x2)2+(y1−y2)2,(x1+x22,y1+y22)
Gradient; forms of a line In the booklet
m=y2−y1x2−x1,y=mx+c,ax+by+d=0,y−y1=m(x−x1)
Perpendicular bisector of AB: through the midpoint of AB, gradient −1mAB Learn it
Voronoi: cell edges lie on perpendicular bisectors of pairs of sites; the point furthest from every site ("toxic waste dump") is at a vertex or on the boundary Learn it

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AI SL formulas →

Voronoi Diagrams worked examples

Worked example 1: Voronoi vertex — three-way boundary · hard

Boundary P–Q: y=2x−1. Boundary Q–R: x+y=14. Find the Voronoi vertex.

Solution

4 steps · 1 to fill in
Examiner tip: A Voronoi vertex is equidistant from all three neighbouring sites.

Worked example 2: Identifying the closest site · easy

Voronoi diagram of hospitals H1,H2,H3. Accident at point P. Which hospital dispatches?

Solution

3 steps
Examiner tip: No distance calc needed — cell boundaries already guarantee the closest site.

Worked example 3: Constructing a Voronoi boundary · medium

Sites A(2,2) and B(6,10). Find the boundary separating them in y=mx+c form.

Solution

4 steps
Examiner tip: Voronoi boundary = perpendicular bisector.

Voronoi Diagrams exam-style questions

3 questions from our AI SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Each has a full mark scheme in the practice area.

Question 1 · easy · 5 marks · Paper 1

The points A(2,1) and B(6,9) are given.

  1. Find the midpoint of [AB]. [1]

  2. Find the gradient of [AB]. [1]

  3. Find the equation of the perpendicular bisector of [AB], in the form ax+by+d=0. [3]

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 1

A city has two major cell phone towers, T1 at (−2,4) and T2 at (6,−2). The Voronoi boundary separating their coverage areas is the perpendicular bisector of the line segment joining them.

Find the equation of this boundary line, giving your answer in the form ax+by+d=0, where a,b,d∈Z.

Full worked solution Free, no sign-in

Midpoint of T1T2: M=(−2+62,4−22)=(2,1) A1
Gradient of T1T2=−2−46−(−2)=−68=−34
Perpendicular gradient (negative reciprocal) =43 M1 A1
Equation through M: y−1=43(x−2)⟹3y−3=4x−8
Rearrange to integer form: 4x−3y−5=0 A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 1

Two locations on a map are represented by the coordinates A(−2,5) and B(4,1).

  1. Calculate the coordinates of the midpoint of [AB].
  2. Find the exact gradient of the line passing through A and B.
  3. Hence, determine the equation of the perpendicular bisector separating A and B. Give your answer in the form y=mx+c.
Attempt it and see the mark scheme →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 5 pages · 247 KB

All 33 Voronoi diagrams questions →

Where students lose marks on Voronoi diagrams

Examiner's noteVoronoi diagrams and bisectors: midpoint, negative reciprocal, vertices

Where marks are lost: Candidates used a site instead of the midpoint, did not take the negative reciprocal, rounded the gradient early, wrote a point or the wrong line instead of the equation of a vertical bisector, found an intersection of given edges instead of the required vertex, and could not find a bearing or a meeting point of two bisectors.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Voronoi Diagrams FAQ

How do I find the equation of a Voronoi edge?

It lies on the perpendicular bisector of the two sites: find their midpoint, take the negative reciprocal of the gradient between them, and use y − y₁ = m(x − x₁) through the midpoint.

What is the toxic waste dump problem?

Finding the point in a region that is as far as possible from all the sites. The answer is at a vertex of the Voronoi diagram or on the boundary of the region, so you compare the distances from those candidate points to their nearest site.

How many IB Maths AI SL Voronoi diagrams questions are there?

There are 33 exam-style Voronoi diagrams questions in the AI SL question bank (Paper 1: 19 · Paper 2: 14), graded 6 very hard, 11 hard, 11 medium, 5 easy. Every question has a full IB-style mark scheme (M, A and R marks).

Is Voronoi diagrams on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 19 · Paper 2: 14. Practise with your GDC — AI papers expect calculator methods throughout.

Next steps

More AI SL Unit 3: Volume and Surface Area · 3D Coordinate Geometry · 3D Trigonometry · Arcs and Sectors · Coordinate Geometry · Non-Right-Angled Trigonometry · Right-Angled Trigonometry · Three-Figure Bearings and Navigation

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Constructing a Voronoi diagram · Nearest-point problems from Voronoi.

Keep revising AI SL: AI SL revision notes · IB Maths AI SL past papers · AI SL common exam mistakes · AI SL syllabus checklist · AI SL geometry and trigonometry knowledge organiser

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