Math Is Fun Forum

circlemaker

Member

Registered: 2011-04-03

Posts: 20

Geometry: Using intersecting circles to obtain square roots

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.

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

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Geometry: Using intersecting circles to obtain square roots

Very cool stuff

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

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

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!!

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igloo myrtilles fourmis

circlemaker

Member

Registered: 2011-04-03

Posts: 20

Re: Geometry: Using intersecting circles to obtain square roots

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.

circlemaker

Member

Registered: 2011-04-03

Posts: 20

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.

philandy

Member

Registered: 2013-06-10

Posts: 11

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?

circlemaker

Member

Registered: 2011-04-03

Posts: 20

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;