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### 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

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/
```
Associated Topics:
College Conic Sections/Circles
High School Conic Sections/Circles

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