Nesting Shapes

Geometry Level 4

I begin with a circle of radius \(1 \text{ cm}\), which becomes the inscribed circle of an equilateral triangle, which I enclose with its circumcircle (the circle that touches all of its vertices). I enclose this circle with a square (so that each edge touches the circle once), and surround this with the square's circumcircle. I repeat this procedure with a pentagon, a hexagon, and so on forever (increasing the side number by one, until we reach an infinity-sided polygon).

What is the limiting area (in \(\text{cm}^2\)) of the "final" circle, rounded to a whole number?

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