Okay, say you're knitting a doily from the center outward and you know how many rounds the whole doily is. It turns out that that there's a fairly straightforward way to calculate when you're halfway done (or whatever). You can't just say "I've knit 30 rounds out of 60, I'm halfway done", because the number of stitches per round keeps growing.
Here's the easy math: take the number of rounds you've knit so far and square that number (multiply it by itself). Take the number of rounds you're going to knit and square that. Divide the former by the latter, and that's the percentage.
30 rounds out of 60 means
(30 x 30)/(60 x 60) = 900/3600 = 25% or a quarter done.
The area of a circle is π times (r squared), where r is the radius of the circle measured in whatever units you're using.
If you think about it, rounds can be used as a unit of measurement. After all, if you're knitting something and you know that your row gauge is 5 stitches per inch, then you can count rows to know how many inches you have.
When you count how many rounds you've knit, you're measuring the radius of a circle, because you're measuring from the center of the circle.
Say you've knit 20 rounds out of 60. The area of the circle you've already knit is 20 rounds squared times pi, or 400π square rounds. ("square rounds" makes me grin, and is a strange unit of measurement--however, it will go away when we calculate the percentage.)
The area of the circle you will be knitting is 60 rounds squared times pi, or 3600π square rounds.
To find out what percentage the smaller circle is of the larger, you divide the area of the smaller by the area of the larger, so:
(400π square rounds)/(3600π square rounds)
Yes, that's a fraction. Fractions being what they are, there's a bunch of stuff we can cancel out: pi divided by pi is 1; square rounds divided by square rounds is 1, leaving us with 400/3600, which is one ninth of the circle, or about 11%.
It turns out, by the way, that you'll be about halfway done knitting any circle (from the center outwards) when you've knit about 7/10 of the rounds (or to be precise, 1/√2 ). In the case of the 60 round doily, that's about 42 rounds.