Radius of an Arch

Date: 05/25/98 at 15:52:07
From: Travis Bullard
Subject: Radius of circle

Is there a simple formula to calculate the radius of a circle where 
the chord length and a distance from the centre of the chord to the 
circumference are known? This is for my dad who is a carpenter. He 
needs it to scribe archways in windows and door openings. 

Thank you for your attention.

Date: 06/06/98 at 12:52:26
From: Doctor Jeremiah
Subject: Re: Radius of circle

Hi Travis:

There is a fairly simple equation to find the radius:

     |-------------L-------------|

                +++++++              ---  ---
           +               +          |    |
        +                     +       d    |
      +                         +     |    |
     +---------------------------+   ---   |
                                           R
                                           |
                                           |
                                           |
                   +  <=== circle center  ---

Here is the information we have:

   L is the Length of the chord length
   d is the distance from the centre of the chord to the circumference

Here is the information we are trying to find:

   R is the Radius of the circle

Let's add a few dimensions to our drawing:

The radius (R) of the circle is the distance from the center to any 
point on the circumference.

We also need the length from the end of the chord to halfway along 
where it crosses the center mark. This is the chord length divided 
by two.

The distance from the center of the circle to the chord is the radius 
of the circle (R) minus the distance from the chord to the edge.

     |-----L/2-----|

                +++++++
           +               +
        +                     +
      +                         +
     +---------------------------+   ---
       +                              |
          +                           |
             R                       R-d
                +                     |
                   +                 ---

If we draw it like this where d and L are known then we have a right 
angle triangle:

                +++++++
           +               +
        +                     +
      +                         +
     +------L/2----+ ------------+
       +           |
          +        |
             R    R-d
                +  |
                   +

The Pythagorean Theorem can be used:

   R^2 = (L/2)^2 + (R-d)^2

Now we just have to solve for R. That means we have to expand (R-d)^2 
by squaring it. Then we simplify and collect terms. The resulting 
equation is quite simple.

If you need more help write me back.

-Doctor Jeremiah,  The Math Forum
Check out our web site! http://mathforum.org/dr.math