Construct a circle of $r=1$. Inscribe a 3-sided regular polygon (an equilateral triangle) inside the circle. Inside the equilateral triangle, inscribe a circle. Inside the second (smaller) circle, inscribe a 4-sided regular polygon (a square). Continue this process infinitely (preferably not by hand). What is the area between the circles and the polygons (where the $nth$ polygon is inscribed in the $nth$ circle and not the other way around)? This is the model after the 2nd iteration:
You would be finding the area in the shaded blue regions after an infinite amount of iterations.
My solution attempt:
$$A_{\textrm{regular polygon of n sides}}=\frac{n}{2}\sin(\frac{2\pi}{n})$$ $$A=(\pi(1)^2-\frac{3}{2}\sin(\frac{2\pi}{3}))+(\pi(\frac{1}{2})^2-\frac{4}{2}\sin(\frac{2\pi}{4}))+...$$
I got stuck here because I couldn't figure out the equation for the radius of the $nth$ polygon or the $nth$ circle.
I apologize for the lack of clarity and some help improving it would be greatly appreciated.