Pi

Archimedes didn't discover π, it was well known that the ratio of the circumference to the diameter of a circle was somewhere about 3 well before Archimedes. It was even recognized that this natural number was important in calculating the area of a circle, by the Egyptian, Celtic, and the Chinese engineers, a millennia or two before the birth of Archimedes.

Now for how they estimated π, they approached the circle as a limit of a regular polygon. They were aware of the properties of triangles, and knew that as a regular polygon got more and more sides it became a circle. The Egyptians exploited this method so well that they a near π that was accurate to four or five decimal places.

The big π discovery; however was during the age of enlightenment (about the 16th century) when π was shown to be irrational.

IMO one of Archimedes great discoveries; however, was his discovery of a method of calculating the volume of a sphere, the screw pump, and hydraulics. But then Archimedes could well have been one of the great minds of the ancient world.

Rap