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**circlemaker****Member**- Registered: 2011-04-03
- Posts: 20

Here's a simple geometric method for obtaining a square root using circles.

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.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,529

Very cool stuff

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,552

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!!

**igloo** **myrtilles** **fourmis**

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**circlemaker****Member**- Registered: 2011-04-03
- Posts: 20

John E. Franklin wrote:

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.

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**circlemaker****Member**- Registered: 2011-04-03
- Posts: 20

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.

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**philandy****Member**- Registered: 2013-06-10
- Posts: 11

circlemaker, that's a beautiful tool. Can you give us a single line of code that turns n into the square root? Can you also show how this can be done on a simple calculator or on paper?

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**circlemaker****Member**- Registered: 2011-04-03
- Posts: 20

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;

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