Which Greek mathematician calculated pi's bounds with polygons?

Archimedes, who trapped pi between two numbers by drawing polygons inside and outside a circle and calculating their perimeters. With 96-sided polygons he established that pi lies between 3+10/71 and 3+1/7, which is correct to two decimal places and was achieved without decimal notation, algebra or a symbol for zero.

It is the method that matters more than the number. He was not measuring but proving - establishing rigorous upper and lower bounds and squeezing, which is essentially integration two thousand years before calculus. He used the same approach on the area of a parabola segment, the volume of a sphere, and the sphere-and-cylinder result he wanted carved on his tomb, which Cicero later found overgrown in Syracuse.

One of his works was lost and then found twice. The Method, in which he explains how he actually discovered results before proving them, survived only in a manuscript that was scraped clean in the thirteenth century and overwritten as a prayer book. It was identified in 1906, disappeared again during the twentieth century, resurfaced at auction in 1998, and was read with X-ray fluorescence and imaging - revealing that Archimedes had been using something very close to the concept of infinity in a way nobody had known.