Once a Voronoi diagram is drawn, the typical exam tasks are: (1) decide which site is closest to a given point, (2) compute the actual distance to that site (often with a scale factor), and (3) use the nearest-site’s data to predict a value for the given point — the technique called nearest-neighbour interpolation. Every question reduces to the distance formula and “smallest wins”.
Most AI SL Voronoi-interpretation problems boil down to three skills, applied in sequence. The flowchart below tells you which calculation to reach for.
If a point falls exactly on an edge, it’s equidistant from the two sites either side. You can use either site’s data for prediction — or quote both as a tied answer.
Worked examples
WE 1Which site is closest?
A Voronoi diagram has three weather stations at A(2, 8), B(7, 7) and C(5, 2). A new farm is built at P(4, 5). Which station should be used to predict rainfall at the farm?
Compute distance from P to each station
d(P, A) = √((4−2)² + (5−8)²) = √(4+9) = √13 ≈ 3.61
d(P, B) = √((4−7)² + (5−7)²) = √(9+4) = √13 ≈ 3.61
d(P, C) = √((4−5)² + (5−2)²) = √(1+9) = √10 ≈ 3.16
Pick the smallest
√10 < √13 → station C is closest
use station C’s data
P is exactly equidistant from A and B (both √13), but C beats both — so cell membership is clear. The “tie” between A and B doesn’t matter because neither is the winner.
WE 2Shortest distance with a scale factor
On a Voronoi diagram, the point P(4, 6) lies in the cell of site M at (8, 3). The scale of the diagram is such that 1 unit represents 4 km. Find the shortest distance from P to its nearest site, giving your answer in km.
P is in cell M → nearest site is M
Distance formula
d = √((4−8)² + (6−3)²)
= √(16 + 9)
= √25 = 5 units
Apply scale factor (1 unit = 4 km)
real distance = 5 × 4 = 20 km
shortest distance = 20 km
(3, 4, 5) Pythagorean triple — gives a clean 5 units on the diagram. Always multiply by the scale at the END, after the distance is found in diagram units.
WE 3Point on a boundary — equidistant from two sites
A delivery hub is located at H(5, 3). Two supermarkets are at X(2, 7) and Y(8, 7). Show that H lies on the boundary between cells X and Y, and state the distance from H to its nearest site.
A point is on the X–Y edge ⇔ HX = HY
Compute HX
HX = √((5−2)² + (3−7)²) = √(9 + 16) = √25 = 5
Compute HY
HY = √((5−8)² + (3−7)²) = √(9 + 16) = √25 = 5
Compare
HX = HY = 5 → H is on edge between X and Y ✓
nearest distance = 5 units (tied: X and Y)
when a point is on an edge, BOTH adjacent sites are tied for “nearest”. The shortest distance is the same number; the prediction can use either site (or quote both).
WE 4Find the vertex of three cells from sites
A Voronoi diagram has three sites A(0, 0), B(8, 0), C(4, 8). The diagram has a single internal vertex V. (a) Find the coordinates of V. (b) Find the distance from V to its nearest site.
(a) Find V — intersect two perpendicular bisectors
⊥AB: AB is horizontal, midpoint (4,0)
⊥AB: x = 4
⊥AC: midpoint (2,4), grad AC = 8/4 = 2
perp gradient = −1/2
⊥AC: y − 4 = −½(x − 2) → y = −x/2 + 5
Intersect: x = 4 → y = −2 + 5 = 3
V = (4, 3)
(b) V is equidistant from all three sites (it’s a vertex)
VA = √(16+9) = √25 = 5
VB = √(16+9) = 5 ✓
VC = √(0+25) = 5 ✓
V = (4, 3), distance to nearest site = 5
a Voronoi vertex is equidistant from THREE sites — so computing one of the three distances is enough. Verifying all three is a good check.
WE 5Nearest-neighbour interpolation
Four fast-food branches and their average daily customer counts: A(2, 5) — 120 customers; B(9, 5) — 180 customers; C(5, 1) — 95 customers; D(5, 9) — 150 customers. A new branch is planned at P(6, 3). Use nearest-neighbour interpolation to estimate the daily customer count at P.
Find the closest branch to P
d(P, A) = √(16+4) = √20 ≈ 4.47
d(P, B) = √(9+4) = √13 ≈ 3.61
d(P, C) = √(1+4) = √5 ≈ 2.24
d(P, D) = √(1+36) = √37 ≈ 6.08
Smallest is √5 → C is closest
Nearest-neighbour interpolation: predicted = C’s value
prediction = 95 customers/day
estimated daily customers ≈ 95
nearest-neighbour interpolation IGNORES the other sites’ values. Only the closest one matters. The assumption is that nearby locations behave similarly — a strong assumption that may be wrong, but it’s the IB AI SL method.
WE 6Multi-step — nearest warehouse, scale, and cost
A delivery company has three warehouses at P(2, 2), Q(8, 2) and R(5, 7). A new customer is at C(5, 5). 1 unit on the diagram represents 10 km, and delivery costs €0.50 per km. (a) Which warehouse should deliver? (b) Find the delivery cost.
(a) Find the closest warehouse to C
d(C, P) = √(9+9) = √18 ≈ 4.24
d(C, Q) = √(9+9) = √18 ≈ 4.24
d(C, R) = √(0+4) = 2 ← smallest
warehouse R is closest
(b) Convert to real distance using scale
real distance = 2 × 10 = 20 km
Multiply by cost rate
cost = 20 × €0.50 = €10
(a) R · (b) €10
three-step combo: nearest-site identification → scale conversion → cost calculation. P and Q are equidistant from C, but R wins, so the tie doesn’t affect the answer.
💡 Top tips
- Always confirm cell membership by distance, not by eyeballing the diagram. Diagrams in exams can be misleading or not-to-scale.
- Apply scale at the END: do all your distance calculations in diagram units, then multiply by the scale factor once.
- Spot Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17). They appear often and give clean integer distances.
- “Nearest neighbour” = nearest only: ignore all other sites once you’ve identified the closest one. The prediction comes from that single site.
- Vertex distances are useful: a vertex is equidistant from three sites, so one distance calculation gives you all three.
⚠ Common mistakes
- Reading the wrong cell from the diagram: always verify with distances. Visual inspection alone can fool you on tight boundaries.
- Forgetting the scale factor: if a question says “1 unit = 4 km” and asks for km, you MUST multiply by 4 at the end.
- Averaging multiple sites for prediction: nearest-neighbour interpolation uses ONLY the closest site. Don’t take an average.
- Confusing edge distance with nearest-site distance: a point’s distance to its nearest site is to the SITE itself (the dot), not to the edge of the cell.
- Square-root mistakes: d² = 25 means d = 5, not 25. Don’t forget the final square root.
Up next: The Toxic Waste Dump Problem. Same diagram, opposite question: find the point that is as FAR as possible from any site. The answer is always one of the vertices, and the technique is the largest empty circle. With this you’ll have everything AI SL asks about Voronoi diagrams.
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