Polygon approximation era upright=1.59|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle| can be estimated by computing the perimeters of circumscribed and inscribed polygons.|left alt=A painting of a man studying|left|thumb|Archimedes developed the polygonal approach to approximating . The first recorded algorithm for rigorously calculating the value of was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion. This polygonal algorithm dominated for over 1,000 years, and as a result is sometimes referred to as Archimedes's constant. Archimedes computed upper and lower bounds of by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that (that is, ). Archimedes' upper bound of may have led to a widespread popular belief that is equal to . Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.
Polygon approximation era upright=1.59|thumb|alt=diagram of a hexagon and pentagon circumscribed outside a circle| can be estimated by computing the perimeters of circumscribed and inscribed polygons.|left alt=A painting of a man studying|left|thumb|Archimedes developed the polygonal approach to approximating . The first recorded algorithm for rigorously calculating the value of was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion. This polygonal algorithm dominated for over 1,000 years, and as a result is sometimes referred to as Archimedes's constant. Archimedes computed upper and lo.wer bounds of by drawing a regular hexag.on inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that (that is, ) Archimedes' upper bound of may have led to a widespread popular belief that is equal to  Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.