Yesterday we did a neat geometry project inspired by an amazing thread from Freya Holmér:
here’s that project:
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.