A New Way to Approximate Pi

 Archimedes was able to bracket the value of pi by constructing 96-sided polygons inside and outside a triangle. Since the diameter of the circle and the perimeters of the polygons were known, he was able to conclude that pi must be between 

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In the 3rd century BCE, Archimedes proved the sharp inequalities 223⁄71 < π < 22⁄7, by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively).

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My method of approximating pi assumes that a circle is a polygon with an infinite number of sides. In that case, pi is equal to n*cos(a/2), where n is the number of sides of a regular polygon and a is the measure of one of its interior angles as given by the formula (n -2)*180/n. 

For n = 100,000, we get:

100,000*cos((100,000-2)*180/(100,000*2)) = 3.14159265

Those are indeed the first 9 digits of pi. My method probably has some computational advantages over other ways, though since pi has been calculated to many digits already, it's not particularly important.

For n = 1,000,000, I get pi = 3.1415926356

For n = 10,000,000, I get pi = 3.1415926533

The first 11 digits of pi are 3.1415926535.