Finding Radius Given the Length of a Chord

Date: 04/09/2006 at 19:50:13
From: Steve
Subject: calculating the radius of a partial circle (arch)

Two points on a circle are 78 units apart and the rise in the center
of these two points to the circle (partial radius) is 12.  How do I
calculate the radius? 

Date: 04/09/2006 at 19:56:42
From: Doctor Vogler
Subject: Re: calculating the radius of a partial circle (arch)

Hi Steve,

Thanks for writing to Dr. Math.  Good question.  Three points 
determine a circle, so if you measure the distance between two points
on the circle (78), 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 78
by 2 to get a = 39), and the second distance is b (12), so that your
picture looks something like this:

              ---------
          ---     |     ---
        /         |b        \
     /            |           \
   /      a       |       a      \
  ---------------- ----------------
                 2*a

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


I calculated this by plotting your three points,

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

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

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

and solving that system of three equations for h, k, and r.  I got 

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

(Note that this works only if the arch is an arc of a circle!  It
won't give you the correct result if you have a parabola or something
else.)

Does this help?

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
  http://mathforum.org/dr.math/