Voronoi Applications (AI HL)
Once a Voronoi diagram exists, the real exam questions ask you to use it - deciding which site "owns" a given point, estimating an unmeasured value by nearest-neighbour interpolation, or finding the best (or worst) location within a region using the toxic waste dump problem. This page focuses purely on those application questions, not on constructing the diagram itself. It's part of the broader Voronoi Diagrams topic.
21 questions on this sub-topic.
Key formula
Covered under IB syllabus reference SL3.6: nearest neighbour interpolation and applications of the "toxic waste dump" problem. The distance formula does the actual comparing; the vertex rule locates the best or worst point in a region.
Distance to a site
\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
In the formula booklet under prior learning. Compute this to every candidate site and compare - the smallest value gives the site whose cell contains the point.
Finding a Voronoi vertex
Solve two bisector equations simultaneously - the intersection is equidistant from all three sites involved, and is where exactly three edges meet in exam questions. It's the point a toxic waste dump problem is looking for.
Not in the formula booklet - simultaneous equationsNeed the full syllabus wording and formula reference table? See Voronoi Diagrams.
Worked examples
Sites are at \(A(0,0)\) and \(B(8,0).\) Which site is the point \(P(3,4)\) closer to?
(a)(i) Find the distance from \(P\) to \(A\).
(a)(ii) State which site \(P\) is closer to.
Worked solution
(a)(i) \(PA = \sqrt{9 + 16}\) M1
\(= 5.\) A1
(a)(ii) \(PB = \sqrt{41} \approx 6.40.\) A1
Since \(PA < PB\), \(P\) is in site A's cell. A1
Hospitals are at \(H_1(1, 2)\), \(H_2(7, 3)\) and \(H_3(4, 8).\) An accident occurs at \(P(5, 4).\) Determine the nearest hospital.
Worked solution
Use the distance formula \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\): M1
\(PH_1 = \sqrt{20} \approx 4.47,\) A1
\(PH_2 = \sqrt5 \approx 2.24,\) A1
\(PH_3 = \sqrt{17} \approx 4.12.\) A1
Smallest is \(PH_2\), so \(H_2\) is nearest. A1
Common mistakes
- Mistaking a Voronoi vertex for a site. The point a toxic waste dump question asks for is a vertex - equidistant from three or more sites where cell boundaries meet - not one of the fixed points you were originally given.
- Judging the nearest site from the picture instead of calculating. Diagrams in these questions aren't always drawn to scale - substitute the coordinates into the distance formula for every candidate site rather than guessing which one looks closest.
- Reporting the vertex itself instead of its distance to the sites. A toxic waste dump answer usually wants a radius or distance value, not just the coordinates of the vertex - re-read what the question is actually asking for before you stop.
Ready to practise properly?
21 Voronoi-application questions, marked instantly like the real exam.
Quick answers
How do I decide which site serves a given point?
Calculate the distance from the point to each candidate site using the distance formula, then compare the values - the smallest distance identifies the site whose Voronoi cell contains that point.
What is the toxic waste dump problem?
It asks you to find the point as far as possible from every site within a bounded region - the answer is always a Voronoi vertex, and the radius of the largest empty circle centred there gives the maximum distance achieved.