How did Archimedes discover pi?

Archimedes found a method for finding good approximations to the ration of the circumference of a circle to the diameter. He did this by inscribing polygons inside the circle and finding how the perimeter changes when the number of sides is doubled.

Archimedes came up with an approximation that is equivalent to saying that pi is between 3 1/7 and 3 10/71. This is quite far from being as accurate as what you showed.