He determined the value of pi very precisely, by starting with a hexagon inscribed in a circle, then a 12-sided polygon, then he kept doubling the number of sides until he got to a 96-gon. A procedure like this is called the 'method of exhaustion', and it looks a lot like what we do nowadays with limits.
I am embarrassed to admit that I couldn't figure out how he did it. (I think I was focusing on area, and that might be harder.) I just found a great video by David Chandler (whose youtube channel is Math Without Borders).
Here's a summary:
Start with a hexagon inscribed in a circle of radius 1 (giving diameter of 2). The perimeter of the hexagon will be 6. This gives a lower bound on pi, which is the ratio of circumference to diameter. We know the circumferences is bigger than this perimeter of 6, so pi is bigger than 6/2 = 3.
If you cut one of the triangles that made the hexagon into two, you get a radius that crosses a side of the hexagon at right angles. You can use the Pythagorean Theorem (twice) to find the new side length. Repeat 3 times and you're at the 96-gon. Archimedes had none of our technology, and little or none of our algebraic symbolism, so the calculations were much harder for him. We can do all this on a spreadsheet, and up comes pi (if you have a column for the perimeter over the diameter). So satisfying!
If this doesn't make sense, watch this lovely video. Thank you, David!
Archimedes did a lot more than find a value for pi! What's your favorite bit of calculus that started out with Archimedes?