A fun – and surprising – statistics project inspired by Freya Holmér’s geometry twitter thread

Yesterday we did a neat geometry project inspired by an amazing thread from Freya Holmér:

given three points, you can always find a circle passing through them all

1. draw lines from a to b to c
2. draw perpendicular bisectors (dashed)
3. the circle center is their intersection point
4. the circle radius is the distance from the center to any of the three points pic.twitter.com/9VceD4N8vG

— Freya Holmér (@FreyaHolmer) October 16, 2020

here’s that project:

https://mikesmathpage.wordpress.com/2020/10/17/using-a-great-twitter-thread-from-freya-holmer-for-a-geometry-project-with-my-younger-son/

Today we are extending that project by trying to find the expected area of the circle when the three points are inside of a unit square.

To start the project we talked through a bit of the geometry that we need to answer the question about the expected area of the circle:

Before jumping into the computer simulation we had to check a few more geometric details – here we talk about using Heron’s formula:

Now my son took 15 min off camera to write a simulation to find the expected value of the area of the circle. Here he walks through the program and we look at several sets of 1,000 trials:

Finally, we finish up with a bit of a surprise – switching to 10,000 trials, we find that the mean still doesn’t seem to converge!

Turns out the expected area of the circle is infinite – that’s why we aren’t seeing the mean in our simulations converge. I think this is a great way to show kids an example where the Central Limit Theorem doesn’t apply.