Last update: July 18 2015

Square roots from intersecting circles


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Square:
Root:

Click & drag your mouse up and down to change the measurement, or enter a number in one of the fields and press Enter.

 

sqrt(n)=cos(asin((n-1)/(n+1)))*((n+1)/2)

 

Finding a square root using trigonometry

The interactive example and equation shown above describe how we can find square roots using intersecting circles and trig functions. A key part of the equation is (n+1)/2 which is the intersecting circle radius.

 

 

Below, some javascript code showing the equation:

n = 5; //number to get square root of
root = Math.cos(Math.asin((n-1)/(n+1)))*((n+1)/2); //2.23606797749979

 


 

History

I discovered this in 2008 while while studying the Vesica piscis.

 

Thanks to Bob Blackmon for reducing the equation further. My original equation (below) is a bit less elegant.

sqrt(n)=cos(asin((((n+1)/2)-1)/((n+1)/2)))*((n+1)/2)