Bounding the Circle

Geometry Level 4

The area of a circle with radius 1 is equal to \(\pi\). One way to determine the value of \(\pi\) is by inscribing and circumscribing regular polygons. For instance, it is obvious from the drawing above that the circle's area is greater than that of the smaller square but smaller than that of the bigger square. This shows that \(2 < \pi < 4\)--using squares we find the value of \(\pi\) within a range of \(4 - 2 = 2\).

In the same way, using a circumscribed and an inscribed regular octagon, we find that \(2.82842 < \pi < 3.31371\), with a range of \(0.48529\).

Now suppose that instead we inscribe and circumscribe regular 16-gons, obtain the approximation \(A_{insc} < \pi < A_{circsc}\). What is the range of this approximation?

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