Chords From Inscribed Polygons

Date: 07/11/2002 at 22:36:31
From: Edmond Clay
Subject: Unknown Chord Known Radius 8 parts

I am a carpenter and would like the formula for deriving the length 
of a chord of a circle with a known radius.  The chord here is 
defined as equal parts of the circle and in this particular case is 
an octagon.  I need to know this in order to cut the blocks (spacers)
that fall between the trusses of a turret that I am building.  Thank
you for your time and effort.

Date: 07/12/2002 at 03:01:48
From: Doctor Jeremiah
Subject: Re: Unknown Chord Known Radius 8 parts

Hi Edmond,

To find the side length of an octagon inscribed in a circle with a
known radius, notice first that the distance from the center to the
corner of the octagon is the same as the radius of the circle.  

This means that the octagon is really eight triangles with three
sides, one of which is the chord length, and two of which are the
radius.

           +------L------+
         +  \           /  +
       +     \         /     +
     +        R       R        +
   +           \     /           +
   |   +        \   /        +   |
   |        +    \a/    +        |
   |              +              |
   |        +    / \    +        |
   |   +        /   \        +   |
   +           /     \           +
     +        /       \        +
       +     /         \     +
         +  /           \  +
           +-------------+

But we don't know L, so we need some other piece of data.  It turns
out that angle 'a' is 360/n, where n is the number of sides.  For an
octagon, of course, n = 8.

We can cut one of the triangles in half,

      +----L/2--+
       \        |
        \       |
         \      |
          R     |
           \    |
            \a/2|
             \  |
              \ |
               \|
                +

and use the definition of the sine of an angle:

 sine of angle = length of opposite side / length of longest side

which translates to 

             sin(a/2) = (L/2) / R

           R sin(a/2) = L/2

          2R sin(a/2) = L

But since an octagon has 8 sides, the value of a would be 360/n, and
we have:

  2R sin( (360/n)/2 ) = L

        2R sin(180/n) = L

So in the case of an octagon with a radius of 10:

        20 sin(180/8) = L

                7.654 = L

Let me know if you need more details.

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

Date: 07/13/2002 at 18:57:13
From: Edmond Clay
Subject: Thank you (Unknown Chord Known Radius 8 parts)

Bingo!  The good news is that I applied the calculation to 
the frieze block cuts and installed away.  Everything fit 
perfectly.  It never ceases to amaze me how a very accurate 
calculation always makes the woodwork look good.  I'm in 
production framing and quite often "fuzzy" math is the "go 
to" method for in the field applications.  I had hoped to 
avoid carrying around a trig table, but I suppose I'll have 
to move my craft to the next level now...Now about building 
the compound arch....Thanks Dr. Math!