Select three points uniformly at random inside of a unit square. What is the expected area of the circle passing through those three points?

This question turns out to have a lot of nice surprises. The first is that exploring the idea of how to find the circle is a great project for kids. The second is that the distribution of circle areas is fascinating.

I started the project today by having the kids explore how to find the inscribed and circumscribed circles of a triangle using paper folding techniques.

My younger son went first showing how to find the incircle:

My olde son went next showing how to find the circumcircle:

With that introduction we went to the whiteboard to talk through the problem that Steve Phelps shared yesterday. I asked the boys to give me their guess about the average area of the circle passing through three random points in the unit square. Their guesses – and reasoning – were really interesting:

Now that we’d talked through some of the introductory ideas in the problem, we talked about how to find the area of a circle passing through three specific points. The fun surprise here is that finding this circle isn’t as hard as it seems initially:

Following the sketch of how to find the circle in the last video, I thought I’d show them a way to find the area of this circle using ideas from coordinate geometry and linear algebra – topics that my younger son and older son have been studying recently. Not everything came to mind right way for the boys, but that’s fine – I wasn’t trying to put them on the spot, but just show them how ideas they are learning about now come into play on this problem:

Finally, we went to the computer to look at the some simulations. The kids noticed almost immediately that the mean of the results was heavily influenced by the maximum area – that’s exactly the idea of “extremistan” that Nassim Taleb talks about!

This project is a great way for kids to explore a statistical sampling problem that doesn’t obey the central limit theorem!

I really love the problem that Phelps posted! It is such a great way to combine fascinating and fundamental ideas from geometry and statistics