Circles around a Larger Circle

Date: 07/26/2003 at 10:38:44
From: Kelly Jones
Subject: Circles

Is there a formula to determine the diameter of several smaller 
circles outlining the circumference of a larger circle? For example, 
if I have a circle with a circumference equalling 10 and I want to 
make 10 smaller circles fit perfectly around this circumference, what 
diameter would these smaller circles have? And what if I only wanted 
to make 8 small circles fit, or any other number?

Date: 07/27/2003 at 17:44:24
From: Doctor Jeremiah
Subject: Re: Circles

Hi Kelly,

Lets say you want n circles of radius r around the outside of a middle 
circle of radius R.

If you draw a line from the center of the middle circle that makes a 
tangent with the outside circle, you end up with this triangle:

                     +++++++
                 ++          ++
              +                  +
            +                      +
       \                  A          
        \  +            +           + ---
         \ +         +  |           +  |
          \+      r     |           +  |
           \   +        r              |
            +           |          +   |
          B  \+         |         +    |
              \  ++     |     ++      r+R
               \     +++++++           |
          +++++ \       |        +++++ |
    +++          \      |              | +++
                  \     |              |
                   \    R              |
                    \   |              |
                     \  |              |
                      \a|              |
                       \|              |
                        +             ---
                        C


a = (360/n)/2 because if it were the full angle between two tangent 
lines it would be 360/n. The tangent line makes a 90-degree angle with 
the radius coming from the center of the outside circle. So we have a 
right triangle that looks like this (just rotated a bit):

                     C
                     +
                    /|
                   /a|
                  /  |
                 /   |
                /    |
               /     |
             R+r     |
             /       |
            /        |
           /         |
          /        90|
         +-----r-----+
        A             B

Where, again, a = (360/n)/2 which, if we simplify it a bit, is the 
same as a = 180/n

Using trigonometry:

  sin(180/n) = r/(R+r)
  (R+r) sin(180/n) = r
  R sin(180/n) + r sin(180/n) = r
  R sin(180/n) = r - r sin(180/n)
  R sin(180/n) = r(1 - sin(180/n))
  R sin(180/n)/(1 - sin(180/n)) = r

So, if you know R (the radius of the middle circle) and n (the number 
of circles around the outside) you can calculate the radius of the 
outside circle.

For example:

If you want n=10 outside circles and the middle circle has a 
circumference of 10, then its radius is R=10/(2 Pi) and you can do 
this:

  R sin(180/n)/(1 - sin(180/n)) = r
  10/(2 Pi) sin(180/10)/(1 - sin(180/10)) = r
  5/Pi  sin(18)/(1 - sin(18)) = r
  r = 0.71

So the outside circles would have to have a diameter of 2*.71 = 1.42

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