Archimedes’ computation of would have to be rated highly on his list of achievements. The value of is the ratio of the circumference of a circle to its diameter. He was able to obtain an upper and lower bound by drawing regular polygons with many sides in and around the circle, and computing their perimeter. Archimedes put the value of between 3 ¹⁰/₇₁ and 3 ¹/₇.

Using the geometric formalism outlined in Euclid’s Elements and in particular the method of exhaustion, Archimedes performed a number of geometric proofs in relation to the calculation of areas and volumes. In doing so, he perfected a method of integration, which enabled him to compute areas, volumes and surfaces of many bodies.

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