Kevin Ryland
Elena Nadolinskaya
Victoria Sun
Circles around a circle proof
This proof’s conjecture is the following: there is a formula that can be used to calculate the
radius of the surrounding circles of a larger circle using the radius of the larger circle and the number of
smaller circles. These circles are arranged below
In this particular model, the radius of the larger circle is 1 and the radii of the smaller circles are .5.
Using trigonometry, the number of circles needed to surround the larger circle completely is calculated
below.
Kevin Ryland
Elena Nadolinskaya
Victoria Sun
Referring to the picture above,
The number of circles can then be calculated by
Now if 10 circles were placed around the circle
Kevin Ryland
Elena Nadolinskaya
Victoria Sun
In reference to the picture on the last page,
When graphed, this result is expected to be around .5 so this is an accurate result. This problem allows
for the derivation of a general formula.
For n circles, the radius R needed to “fit” the circles around a larger, center circle of radius x is:
I have been doing some work with this theorem, and so far the possible issues are as follows:


Domain of tangent function (can’t =
)
An error occurs when tan(x)=1, for example

Negative numbers also seem to occur, I have done my best to check the overall validity of this
theorem but it is a bit puzzling. I have assembled a table of values of R vs.
where x is set
equal to 5 on separate excel sheet. My notes in reference to this sheet will follow my report. If
you wish to graph this data, I later found that it can be modeled by the equation
Another interesting thing I found about this theorem is that when x=R
This means that whenever a circle is surrounded by circles of the same size, exactly 6.775819809 circles
can be placed tangent to one another and to the surrounded circle.
Kevin Ryland
Elena Nadolinskaya
Victoria Sun
Excel Data notes
R
200
150
100
50
0
R
0
50
100
150
200
250
300
350
400
450
-50
-100
-150
-200




“Divide by 0” errors and true 0’s seem to change the sign of the data and are approached
asymptotically. Needless to say, that makes each error and 0 an asymptote.
“Tangent Domain” errors have no effect on the graph whatsoever which therefore leads me to
believe that this error can be overcome using a different formula.
The graph seems to have a period of
This graph was chosen because it is simple and has an indirect relation to the number of circles,
n. This relationship is characterized by the formula

Therefore, a graph of R vs. n can be obtained by the equation

This graph is also quite interesting; it forms a hyperbola with 0 as the asymptote. This graph
makes the most sense to me in relation to the original problem. As n is infinitely small, the
radius of the surrounding circle must be infinitely high in order to surround the center circle.
This is of course impossible until n reaches a certain point which leads me to my next analysis.
n must be greater than 2 in order to surround a center circle. If the radii of the surrounding
circles are expanded out to infinity, then even .00000001 of a circle has an area of infinity and
can therefore count as a whole circle. So 2.0000001 circles is equivalent to 3 circles.
Geometrically, it is easy to then deduce that 3 large enough circles can surround a small circle.

Kevin Ryland
Elena Nadolinskaya
Victoria Sun
Circles around a circle (con’t)
After examining some of the properties of my old formula, I deduced that the disadvantages of
it were linked to the formula’s dependence on trigonometry. So to defeat this major flaw, I
devised an alternative method to estimate the number of circles that would fit around a central
circle in the same fashion as the original proof. First, the original set-up is constructed and a
“washer” is placed intersecting all of the outer circles at their centers as shown below.
r
R+r
R
Therefore, the circumference of the washer can be represented by the equation
If the circumference is divided by the diameter of the smaller circles, the number of smaller
circles that can be fitted can be determined or
Kevin Ryland
Elena Nadolinskaya
Victoria Sun
Which gives an overall equation for the number of smaller circles
Where n is the number of smaller circles, R is the radius of the center circles, and r is the radius
of the surrounding circles.
To prove this technique goes hand in hand with my previous theory, I will show an example
worked with both the “Trig. Theory” and the “Washer Theory.”
Problem: How many circles (exactly) of radius 2 can be placed around a center circle of radius 6
tangent to the center circle and one another.
Trig Theory
Kevin Ryland
Elena Nadolinskaya
Victoria Sun
Washer Theory
2
8
6
Kevin Ryland
Elena Nadolinskaya
Victoria Sun
As can be seen by the problem above, the two answers are very close. They are, however, off
by a small margin which can be explained by looking further into the washer theory. In this
theory, the circumference of the washer is divided by the “diameter” of the surrounding circles
to find the number of circles. When examined though, this diameter is just the portion of the
washer’s parameter that is enclosed within each of the surrounding circles. The effect of this is
shown below.
For low n values
As n approaches
infinity