Q&A #6322

Teachers' Lounge Discussion: Measuring arc length

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From: Loyd

To: Teacher2Teacher Public Discussion
Date: 2012010512:50:22
Subject: Re: Finding the radius of a circle when you only have an arc

On 2004030820:02:12, Philip Wetherington wrote:
>I have a board that is 6 inches wide and 20 inches long. I want to
>an arc in the board that starts at the lower left bottom corner of
>board and runs to the lower right corner of the board. In the middle
>of the board the height of the arc will be 4 inches from the bottom
>the board (midpoint along the bottom of the board). If I know the
>formula for this, I can mark the point 4 inches up from the midpoint
>at the bottom of the board, run down the radius line the "proper
>distance",(this is the unknown I'm looking for), anchor the end of a
>string, run back up the distance of the radius, wrap the string
>a pencil and draw my arc.
>This is not a hypothetical question. I am a woodworker and find
>having to construct such an arc quite often using a time consuming
>"trial and error" method. A formula using the 22 inches and 4 inches
>to find the center of the circle, and thus the radius would be a
>Thank you for your help,  
>Philip R. Wetherington
>Macon, GA

There is an answer or two already posted but they require the radius
to solve the problem.  So, I found a theorem that should help get the
radius of the circle that contains the arc.  "If two chords intersect
in a circle then the products of the segments of the chords are
equal."  So, the diameter of the circle and the 22 inch chord are both
chords that intersect.  

The diameter is 4 + S and the other chord is 22 inches divided into
segments of 11 inches each.  So we multiply:
4xS = 11x11

S= 121/4 so the Diameter of the circle is 4 + 121/4.  Half of that is
the radius.  

I found a similar problem in a high school geometry book under a
chapter called special segments in a circle were a clock maker was
trying to repair a broken escape wheel in an antique clock. The book
reference is:



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