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Finding Radius Given Arc Length and Chord to Arc Height

Date: 08/28/2005 at 02:08:25
From: Eric
Subject: I need the radius for a curved peice of glass

I'm purchasing a curved piece of glass for some furniture.  The curve
(arc) is 60 inches long.  The height (the midpoint of the chord to the
center of the arc) is 11 inches.  I need to know the radius of this
curve so the glass company can make my glass.  Any thoughts?


Date: 08/28/2005 at 11:14:48
From: Doctor Vogler
Subject: Re: I need the radius for a curved peice of glass

Hi Eric,

Thanks for writing to Dr. Math.  Someone else asked a similar question
recently.  He took two measurements of the circle (a chord length, and
the length of the perpendicular bisector from the chord to the circle)
and wanted a formula for the radius.  This is what I told him:

Three points determine a circle, so if you measure the distance 
between two points on the circle, then go perpendicular to that line 
from the center of the line and measure the distance to the third 
point on the circle, you can calculate the radius.  If the first 
distance is 2*a (divide it by 2 to get a), and the second distance is 
b, so that your picture looks something like this:

          ---     |     ---
        /         |b        \
     /            |           \
   /      a       |       a      \
  ---------------- ----------------

Then the radius is r = (a^2 + b^2)/(2b).

I calculated this by putting the intersection of the chord and line at
(0,0) and plotting your three points as,

  (-a, 0), (a, 0), and (0, b).

And then plugging those into the general form of the equation for a
circle, namely

  (x - h)^2 + (y - k)^2 = r^2.

I got h = 0, k = (b^2 - a^2)/(2b), and r = (a^2 + b^2)/(2b).

So, Eric, you could use this technique.  If you don't have the length
a (or 2a) but you have the length of the curved side (marked c here)

           c  ---------
          ---     |     ---
        /         |b        \
     /            |           \
   /              |              \
  ---------------- ----------------

then you have a much harder equation to solve for r:

  2br = (sin [c/(2r)])^2 + b^2.

You would probably have to solve this using numerical methods.  So I
would recommend the other idea (and it's probably an easier
measurement to make anyway).

But be warned that these computations assume that your curve is a
portion of a circle.  If it is some other curve, then there is no radius.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum 
Associated Topics:
High School Conic Sections/Circles

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