Circle Radius from Chord Length and Depth

Date: 06/16/99 at 02:18:52
From: Jacques Duranleau
Subject: Finding Radius of circle knowing only chord and depth.

How do you find the radius or diameter of a circle when you only know 
a chord length and the depth? In this case, the chord does not pass 
through the center.

Date: 06/16/99 at 08:45:46
From: Doctor Peterson
Subject: Re: Finding Radius of circle knowing only chord and depth.

Hi, Jacques.

You can almost find this formula in our FAQ on circles. On this page, 
scroll down to Segment of a Circle:

  http://mathforum.org/dr.math/faq/formulas/faq.circle.html   

Unfortunately, it's not quite there, though you could work it out from 
what is there, namely

     c = ... = 2 sqrt[h(2r-h)]

You have h (the height of the chord) and c (the length of the chord), 
and need to find r, so you can solve this for r as

     r = (c^2/(4h) + h)/2

         c^2 + 4 h^2
       = -----------
             8h

You can work out the formula this way:


                     ***********
                *****           *****
            ****                  /| ****
          **                     / |     **
         *                      /  |       *
       **                    r /   |        **
      *                       /    |c/2       *
     *                       /     |           *
     *                      /      |           *
    *                      /  r-h  |      h     *
    *                     *--------+------------*
    *                      \       |            *
     *                      \      |           *
     *                       \     |           *
      *                       \    |          *
       **                      \   |        **
         *                      \  |       *
          **                     \ |     **
            ****                  \| ****
                *****           *****
                     ***********


We have a right triangle, and Pythagoras gives:

     r^2 = (r-h)^2 + (c/2)^2

     r^2 = r^2 - 2rh + h^2 + c^2/4

     2rh = h^2 + c^2/4

           h^2 + c^2/4   4h^2 + c^2
       r = ----------- = ----------
               2h            8h

I hope that meets your needs.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/