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;

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.

]]>The even and the odd ones are

different the way they intersect.

A perhaps more "proper" version would use a circle in the middle for the odd numbers as well, but I didn't like having half circles on the sides.

]]>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!!]]>

1: Start with a circle.

2: Duplicate & divide by the number you want to square

3: Make two intersecting circles which touch the inside of the outer circle and the outside of the inner circle.

4: Where they intersect is the square root.