Normal polygon area without trig functions

[guss]

Here's an interesting problem: How can you find the area of any normal polygon with x sides (or corners) that is inscribed in a circle of radius 1? No trig functions, or things like e or π (Pi), or infinite series, are allowed. If possible, try to avoid summation notation as well, but that might be required.

Normally, you could just break the normal polygon up into triangles, and add the area of each one of those. But, of course, this would use trig functions (at least the way I'm thinking of).

Any way to do this?

This would be especially cool because as x approaches infinity, the area would approach pi.

 

[gb7nash]

Well, we know that the area of a regular n-gon inside of a circle of radius 1 is:

[tex]\frac{n}{2}sin(\frac{2\pi}{n})[/tex]

I can't think of anyway to take away the trigonometry from this, unless you want to allow taylor series. Triangulation is a good idea, but unfortunately, it becomes complicated if you're dealing with an arbitrary n-gon. You would need an algorithm for determining how to slice the polygon into triangles.

I also looked on http://en.wikipedia.org/wiki/Triangle and you can use line integrals to compute the area of an arbitrary polygon. I don't know too much about this method though.

 

[guss]

I was thinking of dividing each shape up into a bunch of right triangles.