## Is there an equation to describe regular polygons?

So a blog reader, Michael Gmirkin, sent me an in-depth email about the possibility of the existence of a super equation that can describe any regular polygon. I wasn’t sure. For reference, you might want to check out these 2 blog posts about the equation for a square: question, answer.

I was going to just ask you here. Then I remembered there’s a Stack Overflow equivalent for maths. So I went there and asked the question. So if you know the answer, you can comment here, or go to the maths StackExchange site and earn yourself some points.

You can assume that the centre of the regular polygon in at the origin (0,0). Researching a little on the topic, I also learnt about the apothem, which is also the shortest distance from the centre to a polygon’s side. The “normal” radius is the distance from the centre to one of the regular polygon’s vertex.

If you trace 2 circles, one with the apothem and one with the radius, you get an inscribed circle and circumscribed circle respectively.