Several years ago, I explored approximating Pi with polygons made of Geometiles. I described my findings in this blog. Fast forward to 2022, when Charles Fleischer showed me a polyhedral torus I’ve never seen before, and its boundary was a regular 30-sided polygon. The fancy name for a 30 sided polygon is *triacontagon.* Triaconta τριάκοντα (triaconta) means ’30’ in Greek, and γωνία (gōnía) means ‘corner’ or ‘angle’. I just *had* to get my measuring tape on this one! But first, I had to check if it was flat. Sometimes it’s hard to tell from a picture, especially one taken with a mobile phone camera.

What I was interested in is the interior angle of a regular 30-gon. It is the angle in the picture below:

The formula for this angle is derived here. It tells us that the interior angle of a regular n-gon is

In our case, *n* = 30, so the interior angle is

Now, the regular pentagon has an interior angle of 108^{o}, and the equilateral triangle has an interior angle of 60^{o}. Adding them up we get 168^{o}, which is exactly what we need to get a regular 30-gon! This is illustrated in the picture below.

We can finally determine how close to the value of π this 30 got gives us. Recall that π is the ratio of a circle’s circumference to its diameter. We measure the circumference of our 30-gon with a ribbon, and then measure the length of the ribbon, which turns out to be** 73 inches **long. The diameter is measured with a tape measure to be **23.125 inches**.

So the approximation of π that we get from this torus is π≈73/23.125=3.16. Not bad at all!

You can build this structure and a myriad of others with a Geometiles Jumbo Set.