Beautiful graphical calculator of circles.  I tried many numbers from 1 to 1000,
including a few with halves thrown in.  The even and the odd ones are
different the way they intersect.  This is really fun to try to memorize!!

Just popping by to update with an equation:


If I didn't write the equation properly or if you can simplify, please do.

Even though I discovered this particular geometric relationship in 2008, I had to learn trigonometry before I could figure out the proper equation.  Also, I'm still learning how to write proper math notation.  I usually do math using javascript functions or by using pure geometry so this is a bit new to me.

You can put this up in the browser url bar: javascript:alert(Math.cos(Math.asin((((n+1)/2)-1)/((n+1)/2)))*((n+1)/2));
Just replace n with the number you want the square root of.
Note: some browsers take off the "javascript:" part when copying/pasting.

The equation is much easier if you define (n+1)/2 separately which is the intersecting circle radius:
n = 9; //get square root of this number
ic = (n+1)/2; //intersecting circle radius
sqrt = Math.cos(Math.asin((ic-1)/ic))*ic;