InterMath
Title
Packing Circles
"Packing" refers to filling the inside of a square with as many circles that do not overlap the
square or any of the other circles. (Assume all the circles have the same length radii.) How
does the ratio of the length of the side of the square and the diameter of the circle compare
with the number of circles that you can pack inside the square?
Problem setup
I am trying to place circles inside a square so that the circles do not overlap with the sides of the
square or with the other circles. I am seeking to find a relationship between the ratio of the
side’s length to the diameter of a circle and the number of circles inside the square.
This problem makes me think about packing Coke bottles in a carton or a case. When packing
items in an enclosed space and wanting to get the best use of the space, it would be important to
know a ratio that could help you utilize the area.
Plans to Solve/Investigate the Problem
Prediction: I predict that the ratio will be more than double the number of circles that will fit
inside the square.
I plan to construct squares of various sizes. I will use the length of the sides to determine how
many circles I will put in my square. I will record the ratio of the length of the side of the square
to the diameter of one of the circles. I will use this number to help find a relationship with the
number of circles I was able to include in my square.
To show consistency with my results, I plan to try different size squares and to adjust the size
circles included in each square.
Investigation/Exploration of the Problem
I began in GSP by constructing two sets of perpendicular lines so that I could connect points to
form perpendicular segments. I adjusted the position of each point so that each segment was
four centimeters. Then, I began constructing circles – each with a diameter of 2 centimeters
(trying to emulate the picture for the task). I soon realized that this could eventually lead me to a
conclusion, but it would take an extreme amount of time. I began using scratch paper to work
through similar problems and started seeing a connection between the numbers.
m DB = 4.00 cm
m BE = 4.00 cm
m EF = 4.00 cm
m FD = 4.00 cm
A
D
K
B
L
M
m KL = 1.01 cm
m KM = 2.01 cm
m GH = 2.01 cm
H
F
G
E
I decided to try Excel to assist me with the task. I created four columns – “length of side,”
“diameter of each circle,” “ratio of length to diameter,” and “number of circles inside the
square.”
On my first chart, I inserted lengths of 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14. Thinking of the
picture, I put four circles inside the squares, so the diameters were half of the length of each side.
The ratio of length to diameter was 2 for each example and the number of circles inside the
square was 4. This held true for all the examples but was not enough information to see a
relationship for all circles (not just those that with diameters that were half the length of the
sides). (see chart below)
length of sides
4
5
6
7
8
9
10
11
12
13
14
diameter of each circle
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
ratio of side length to
diameter
2
2
2
2
2
2
2
2
2
2
2
# of circles in the
square
4
4
4
4
4
4
4
4
4
4
4
To take the task further, I decided to use the same chart but try different sized circles. I started
again with a square with lengths of 4 on each side. Rather than having four circles with
diameters of 2, I chose to have circles with diameters of 1. The ratio for these numbers became
4:1, or 4. I drew a sketch of this on my scratch paper and found that there were 16 circles inside
the square. The numbers 4 and 16 stuck out to me.
I decided to try this again with different numbers. I used a square with sides of 8. I decided to
let the circles have diameters of 1. The ratio was 8:1, or 8. The number of circles inside the
square was 64. Believing I was really onto something, I kept my same square but adjusted the
diameter to 2. This gave me a ratio of 8:2 or 4:1. Just as I suspecting, the number of circles
inside the square was 16. I decided to use formulas in my chart so that any values would give
me accurate results. (see chart below)
length of side
4
6
8
8
10
12
diameter of each circle
1
2
2
1
2
1
ratio of length to diameter
4
3
4
8
5
12
# of circles in the square
16
9
16
64
25
144
In conclusion, when you square the ratio of the length to diameter, you discover the number of
circles that will fit inside the given square.
Extensions of the Problem
I would be interested to know is there a relationship between the diameter of the billiards balls
and the triangular device used to rack the ball?
GPS connections (for 6th and 7th grade)
.
Author & Contact
Erin Lee Hutto
Link(s) to resources, references, lesson plans, and/or other materials