As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for how to convert this into non-trig form. I know it will involve some kind of infinite summation but I have no clue how to work it out. The n-gon is inscribed in a circle of radius b (say). Please don't use b=1 if you help me because I want to see how b enters into the formula.
Thanks
On
Since the formula uses the sin function, it has to be there in some form.
If you want to compute the area of a 2n-gon in terms of the area of a n-gon, you can use the method of Archimedes, which is described very nicely in chapter 38 of "100 Great Problems of Elementary Mathematics" by Heinrich Dorrie ($13 from Amazon, also available online from Google Play Books). I highly recommend this book.
Archimedes' method uses pure geometry to go from the sides of inscribed and circumscribed n-gons to the corresponding sides of 2n-gons. Though the method shown in this book is used to get upper and lower bounds for the circumference of a circle, it could be readily modified to get bounds for the area.
Area of regular $n$-gon in terms of circumradius $b$ is $$A=\frac{1}{2} nb^2 \sin\left(\frac{2\pi}{n}\right),$$so to avoid the sine function expand in a taylor series: $$A=\frac{1}{2} nb^2\sum_{k=0}^\infty \frac{(-1)^k\left(\frac{2\pi}{n}\right)^{1+2k}}{(1+2k)!}.$$