Simon is working on a project that will involve constructing the Archimedean solids from paper pieces that he programs in Processing (Java) and prints out. In the previous video, Simon worked out the distance between two points to measure the side length of a pentagon that has the radius of 1 (i.e. the distance between its adjacent vertices if the distance from its center to its vertices is 1). He first made a mistake in his calculation and got a result that would be true for a hexagon, not a pentagon. He then corrected himself and got the value that he thought he could use in the Processing code, but as it turned out, the ratio between the radius and the side length was still not right. We recorded a whole new video full of calculations and playing with the code, and achieved pretty neat results after Simon used the new value in the code, but still not good enough, as Simon wanted to have his pentagons to have the side length of 40 (to match the triangles and the squares he’d already made). Simon later found a solution using a different formula for a polygon with n sides (from trigonometry, defining the radius as the side length over (2sin times 180/n)) and succeeded in getting exactly the pentagons he wanted, with the side equalling 40. See the result here:

https://www.youtube.com/watch?v=f4unEptU3Vs&t=1s

The winning formula:

If you are really into working out the calculations, feel free to check out our frantic attempts here: