I Discovered a New Shape July 25, 2015Posted by Sobek in News.
Well, it’s new to me, anyway.
I was working through some trigonometry problems, and there were a whole bunch that boiled down to “you have a seven-sided polygon, each side is x cm, find the area of the polygon.” I didn’t do very many of those problems before I decided to make a short cut and just work out a formula:
A=(n [d/2]^2)/tan (180/n)
Where n is the number of sides, and d is the length of one side.
Figuring out that formula saved me some time, but as far as I know isn’t super useful.
At one point, I wondered what would happen if I did some weird stuff with the formula. Like what’s the area of a polygon with two sides? That produces a zero in the denominator, so it’s undefined. That makes sense. But what about a polygon with 1.9 sides? That actually does produce an answer, and so does 1.8. If you set the side length to some constant (I picked 2 so the calculations would be easier), you end up with a dampened tangent/cotangent curve. Even more interesting is that you get holes in the curve at regular intervals.
After all that, I don’t know how to draw a regular polygon with 1.9 sides, but I can tell you how much area it has.