I linked to this method in a post about how to calculate square roots by hand.
But … in the comments was a brand new method:
Sometime in 2008 while I was studying the vesica piscis, I discovered that square roots can be found using intersecting circles.
The cute thing about this is that you could do this to estimate a square root without pencil, paper or calculator … just shapes and a ruler:
- Start with a circle with a radius of 1 [centered on the origin].
- Duplicate it and multiply it's size by the number you want the square root of. [draw another circle, also centered on the origin, whose radius is the number you want a square root of]
- Create two more circles which wrap around the inner circle while connecting to the perimeter of the outer circle. [these new circles should touch the left outside of the small circle and the right inside of the big circle, and the right outside of the small circle and the left inside of the big circle]
- Where these two circles intersect is the square root. [the coordinate on the vertical axis of their intersection]
And, if you’re not interested in doing the estimate by hand, the discoverer includes a formula based on trig functions.
P.S. I had no clue what a vesica piscis is either. It’s the lens-shape you get when you have two circles intersecting, but only when each circle goes through the center of the other one. It notes that the ratio of length to width is the square root of 3, but the method above will get you any square root you like.