In short, how did Archimedes prove that the ratio of the circumference of a circle to its diameter is constant? It’s very easy to do with analytic geometry and calculus, but he didn’t have those tools available to him.
From : http://www.ualr.edu/~lasmoller/pi.html
The first theoretical calculation of a value of pi was that of Archimedes of Syracuse (287212 BCE), one of the most brilliant mathematicians of the ancient world. Archimedes worked out that 223/71 < Pi < 22/7. Archimedes’s results rested upon approximating the area of a circle based on the area of a regular polygon inscribed within the circle and the area of a regular polygon within which the circle was circumscribed.
Beginning with a hexagon, he worked all the way up to a ploygon with 96 sides!
So he didn’t know that it was a constant?
When was that first discovered?
Archimedes contribution wasn’t the proof of the constanct, the knowledge there was a constant preceeded him. It was his discovery, shown above, how one could repeat a process indefinately to get an ever increasingly accurate value for pi.
So who did prove it was a constant?
Yes, certainly it was known in preArchimedean times that the ratio of the circumference of a circle to its diameter is a constant. Eudoxus of Cnidus in the fourth century BCE proved that the area of a circle is a constant times the square of the diameter; the constant relationship of the circumference to the diameter was also wellknown.
I think I recall hearing a couple years ago that someone had found an OldBabylonian (second millennium BCE) cuneiform tablet with an approximate value of pi on it. Ancient Egyptian and Chinese texts also preserve (cruder) values of pi. The knowledge that the circumference had the same ratio to the diameter in every circle was old, old news to Archimedes. I don’t know where the first known proof of it is, though.
Everything you ever wanted to know about how Archimedes proved his result is here.
Everything you wanted to know about pi, but were afraid to ask.
bibliophage’s article is nice and all that, but I’m really interested in the proofs of three facts using no calculus:

That the area of a circle is a constant multiple of the square of its radius (got that one, thanks to Kimstu).

That the circumference of a circle is a constant multiple of the radius.

And that those two constants are the same.
Well, one of Archimedes’ theorems was that the area of a circle is half the radius multiplied by the circumference. More specifically, in Heath’s translation, Proposition 1 of Measurement of a Circle is:
Following from the earlier result cited by Kimstu, one can then define pi via C=2 pi r. If A = rC/2 by the above theorem, then A = pi r[sup]2[/sup].
At a glance, Archimedes’ proof of the theorem doesn’t seem to involve even an anticipation of calculus.
What’s the proof that the circumference of a square is a constant multiple of its diagonal, or the analogous thing for other polygons? Once you’ve got that, the circle thing seems pretty clear (I know that’s not a formal proof).
You just slice the circle up into equalsized wedges and arrange them in a rectanglelike thing. As the number of wedges gets large, this looks more and more like a rectangle with a width of r and a length of half the circumference.
Take the limit, eh? That’s calculus.
Actually, it does. Archimedes didn’t have the concept of a limit predefined for him, but he used it nonetheless in this and several others of his proofs. Basically, he had to rederive the concept each time he used it, which had to be cumbersome for him.
Basically, his proof amounted to “Given any value less than pir[sup]2[/sup], I can inscribe a polygon with area greater than that value, so the area of the circle is not less than pir[sup]2[/sup]. And given any value greater than pir[sup]2[/sup], I can circumcribe a polygon with area less than that value, so the area of the circle is not greater than pir[sup]2[/sup]. Since the area of the circle is neither greater than nor less than pir[sup]2[/sup], it must be equal to pir[sup]2[/sup].”.
At a second glance, agreed.
Call it what you like. It doesn’t involve an integral (which is what comes to mind when I think of finding areas using “calculus”). It doesn’t even involve taking the limit of a specified expression. One needs no exposure to calculus to understand it. I think people were doing this sort of thing long before the advent of calculus as we know it. If you want to say that they were really doing calculus but didn’t know it, fine, but be prepared for someone to say that they were really doing something else even more advanced (I don’t know, maybe measure theory).