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•  » Geometry: Using intersecting circles to obtain square roots

## #1 2011-04-07 13:30:58

circlemaker
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### Geometry: Using intersecting circles to obtain square roots

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

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.

http://www.perspectiveinfinity.com/Circ … roots.html

## #2 2011-04-08 07:07:57

MathsIsFun

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### Re: Geometry: Using intersecting circles to obtain square roots

Very cool stuff

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

## #3 2011-04-27 08:33:08

John E. Franklin
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### Re: Geometry: Using intersecting circles to obtain square roots

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

## #4 2011-04-28 13:47:47

circlemaker
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### Re: Geometry: Using intersecting circles to obtain square roots

#### 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.

## #5 2013-06-09 04:39:51

circlemaker
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### Re: Geometry: Using intersecting circles to obtain square roots

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.

## #6 2013-06-11 16:22:40

philandy
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### Re: Geometry: Using intersecting circles to obtain square roots

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?

## #7 2013-06-11 17:38:55

circlemaker
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### Re: Geometry: Using intersecting circles to obtain square roots

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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