Circles and Square Roots

Here's a neat geometry trick that you can do at home.

Start by drawing two lines: the axis, on top, and the intercept, on the bottom. Here I've chosen to make the lines 5 units long, but any length will do. The axis and the intercept should be one unit apart.

Draw the perpendicular bisector from the axis to the intercept, and draw a circle through those two points:

Now, do this process again, drawing another bisect and another circular arc, as shown below.

Do it again.

Do it one more time.

That's interesting. Try doing it five more times.

Neat. Seven more times?

How about nine more times now?

If you actually calculate the intersection points of the axis, you will find that they meet the axis at the square roots of the natural numbers!

Why does this work? The first step initializes the process at 1, and each successive step constructs a right triangle with one side being the previous length, and another side being one unit long. By the Pythagorean theorem, sqrt(n)² + 1² = sqrt(n+1)².

I found it interesting that you can use circles and straight lines to find the square root function.