#MathArtChallenge 89: Voronoi diagrams

The Challenge: Throw some random points (or carefully selected ones!) on a plane. Identify the parts of the plane that are closest to each of those points.

Voronoi Diagram Sketched in Notebook

I made mine (above and below) using perpendicular bisectors between the points shown, but are there other ways? How might you reason your way into making one of these? (Hint, start with just 2 points, then 3, then…)

Part of my reason for selecting these is their versatility for the classroom. There are so many ways to use them for deep mathematical discussion that can leverage your student’s knowledge. In the map below, I marked the large Minneapolis Public High Schools with a star and then created a Voronoi diagram. THINK of the conversations you could have about this map.



  • Which high school, just based on this map, do you think is the largest?
  • Are the high schools reasonably distributed throughout the city?
  • Which high school is easiest for the most people to get to? Which is the hardest? Just THINK about what your kids could bring to this discussion. Their knowledge of roads, bus maps (MPS students use the public bus system for high school), safety, residency.
  • If we were to add a new high school, what area makes most sense to place it in?
  • Where do you think students are most densely concentrated?
  • How else might we make this map? Would it make sense to instead base the lines on a street grid map? (So you’re closest according to driving distance, not how birds fly?) Or based on population?
  • What are the benefits of true community schools (defined here as “closest school to you”)? What are the drawbacks?
  • If we laid a map of the elementary and middle schools over this, what would you expect to see? What would you want to see?

And naturally, there are easy ways to make these digitally. Here’s a quick version made in Geogebra:

Author: Annie Perkins

Math Teacher in Minneapolis, MN.

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