Radius from an Arc and a Chord

Date: 06/08/99 at 19:38:11
From: Jackie Castellano
Subject: Geometry, arcs

If I know the height of an arc from the midpoint on a chord, and the 
length of the chord, can I find the radius of the circle of which the 
arc is a part? I can't find a formula.

Date: 06/09/99 at 12:11:16
From: Doctor Peterson
Subject: Re: Geometry, arcs

Hi, Jackie.

You could work out this formula from what is in our FAQ on Circle 
Formulas:

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

Look down the page for segments of circles, namely

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

You're given h and c, 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/