It's nice that you asked about the Greeks, because it was the Greek Archimedes who was the first, so far as we know, to develop a systematic way to calculate pi to as much accurcy as needed. Before that various people had made estimates essentially based on measurement.
How would you measure pi? One way would be to take a large number of little sticks and make a circle out of them. You could then divide the number of sticks around the circumference by the number in a diameter. You could also do something like that with lengths of string.
Archimedes instead tried drawing a series of polygons with more and more sides. I believe he started with a hexagon, whose perimeter has a length of exactly 6 when inscribed in a circle of diameter 2. That gives a starting estimate for pi of 3. Then he made a 12-sided polygon by connecting the vertices of the hexagon to the midpoints of the arcs between them. He used trigonometry and algebra to calulate exactly the perimeter of this dodecagon (12 sides). He repeated the procedure for 24, 48, and 96 sides. Each time the perimeter can be calculated exactly, and each is a little larger than the previous one. In principle this process can be continued forever, getting figures closer and closer to a circle, with perimeters closer and closer to 2pi.
You can also circumscribe the polygons -- drawing one inside the circle and one outside the circle. That way, you get an estimate not only of pi but an estimate of the maximum amount it can be wrong; one polygon will overestimate it and the other will underestimate it.
(published on 10/22/2007)