“Formula in Finding the Area of An Elastic Band Holding Four Circles”
MATHEMATICS INVESTIGATION PROJECT
M3112 Problem Solving, Mathematical Investigations, and Modeling
An Investigatory Paper Presented to:
MILDIN J. RETUTAS Ph.D.
M3112 Instructor
Prepared by:
RHEA JEANNETE S. AVANCEÑA
JAYLYKA MAE B. GONO
JINA FLOR A. PAMONAG
BSED-MATH 3SM
JANUARY 3, 2024
ABSTRACTION
INTRODUCTION
Elastic bands, often called rubber bands, are practical loops made of pliable
rubber that are used to fasten items together (Merriam-Webster, n.d.). They serve an
essential purpose of holding numerous items in place in daily life, and their innate
flexibility and stretch ability make them indispensable. Actually, its application is not
inherently linked to mathematical or geometrical factors in the context of geometry.
The use of elastic bands is more commonly focused on their ability to secure objects
than on using them as a tool for investigating geometry or mathematics.
One fascinating feature of elastic bands is how they can change an object by
wrapping it in them. The elastic band has no distinct shape until it is constricted around
an object; it conforms to the curves of the object it is enclosing. Elastic bands' ability
to adapt and conform to various shapes and sizes is a vital aspect of their overall
functionality. The ability of a material to return to its original shape after deformation is
known as its degree of elasticity, and this attribute is essential to comprehending the
behavior of elastic bands (Theory of rubber conformation, n.d.).
In this investigation, we go beyond the real-world uses of elastic bands to
examine the complex mathematical principles that underlie their behavior. In
particular, we want to find a formula for figuring out how much space there is inside an
elastic band that contains four circles of the same size. Although techniques and
equations have been created to calculate the circumference of elastic bands, there is
still a significant understanding gap regarding the band's inside, which consists of four
identical circles.
By investigating and clarifying the mathematical equations underlying the inside
structure of elastic bands supporting four circles, our research aims to close this gap.
In doing so, we hope to derive a simpler formula on the mathematical properties of
elastic bands and go beyond their traditional practical applications.
REVIEW OF RELATED LITERATURES
This chapter includes the essential literatures, theories, and existing
mathematics that the investigators have applied in order to come up with a new
formula determining the area covered by an elastic band. The literatures provided will
give insights and prior knowledge to the formulation of this investigation.
Elastic Band
In math, when circles sit neatly inside a square, touching all its sides, we often
imagine the square as an elastic band that can stretch and shrink to fit any size circle
while keeping it snug. This "elastic band" analogy is especially helpful when studying
how changes in the circle sizes affect the overall arrangement, revealing the flexible
and dynamic nature of this geometric relationship.
Area of a Circle
A circle is a shape containing a set of points that are equidistant from a given
point. The point is called the center, and the distance from the center to any point in
the circle is called the radius. Twice the length of the radius is called the diameter –
it’s the line segment that passes through the center and has its endpoints on the circle
itself. Lastly, the distance around a circle is called the circumference, which is called
a perimeter in a polygon. The area is the space enclosed inside the circle. It’s
measured in square units such as 𝑚𝑚 2 , 𝑐𝑚 2, 𝑚 2 , 𝑘𝑚 2 . To find the area of a circle,
multiply the pi () which is approximately 3.14 by the squared radius 𝑟 2 .
Area of a Square
A square is a shape with four equal sides (equilateral) and four equal angles
(equiangular). It’s the only equilateral and equiangular quadrilateral. The area of a
shape refers to the total amount of space enclosed in it. To find the area of a square,
multiply Base by Height, as expressed in the formula A = BH; however, since a square
has equal sides, we can write the formula as: A= (s)(s), where: s = length of one side
of the square.
Area of a Rectangle
A rectangle is a four-sided shape with equal and parallel opposite sides.
Remember that area refers to the total amount of space enclosed in a shape. The area
is expressed in square units such as square meters, square inches, or square
kilometers (km2). To find the area of a rectangle, multiply its length by its width, as
expressed in the formula: A=L x W, where L = length of the rectangle; W = width of
the rectangle.
METHODOLOGY
When determining the area of an elastic band, there are a variety of techniques
available. One approach involves dissecting the objects into sections, calculating the
area of each section independently, and then combining it with the gaps to arrive at
the total area. We used a simple method by simply add all the areas of the circles and
the gaps that looked like half of the diamond together. This method was inspired by
Finding the Area of Composite Shapes. The figure below illustrates the area covered
by an elastic band wrapping four circles and the possible formations it can create.
Figure 1.
Area = (Area of
) + (Area of
)
We treated the shape formed by the elastic band as a composite shape, and
by adding the area of the shape circle with the area of gaps that resembles half a
diamond, one could be able to find the area covered by an elastic band. However, it
is crucial to ensure that we measure the radii of the four circles with great accuracy,
as even slight variations can significantly impact the calculated area.
Lastly, finding the area of the shape diamond is not as easy in finding the area
of a square. We also have employed the technique of subtracting the areas of two
objects in order to find the area of these diamond gaps between the circles and the
elastic band.
DISCUSSION
To arrive with the formula, we made sure that the radius of the circles is
predetermined as it is the only measurement we can base on. The formula is derived
when the height or the radius is given. The following are the illustration of how the
formula is derived and the latter part is composed of examples applying the formula.
Illustration:
Step 1. Understand the Problem
•
Understand that in order to find the area, we need to combine all the areas of
circles and the gaps.
Step 2. Devise a Plan
•
To devise a plan, we need to determine how to find the area of the diamond
shape. In finding its area, it requires another solution, here are the two ways…
x
Area of
= Area of Square − Area of
Area of
= Area of Square − Area of a Circle
Area of
= s2 − πr 2
Let r = radius; s = 2r
•
To find the area of
Area of
= (2r)2 − πr 2
𝐀𝐫𝐞𝐚 𝐨𝐟
= 𝟒𝐫 𝟐 − 𝛑𝐫 𝟐
take half of the area of \
Area of
Area of
1
(4r 2 − πr 2 )
2
1
= 2r 2 − πr 2
2
=
Multiply by 4, since there are four
1
= 4(2r 2 − πr 2 )
2
= 𝟖𝐫 𝟐 − 𝟐𝛑𝐫 𝟐
𝐀𝐫𝐞𝐚 𝐨𝐟
We can also utilize another way in finding this area but still yield the same result.
By drawing a rectangle out from the area where the
gaps are located as shown in the figure, we can have
an illustration of a semicircle inscribed in a rectangle.
r
r
r
r
2r
So, the other way could be…
Area of
= Area of rectangle − Area of a semicircle
Let r = radius; length = 2r; width = r
Area of
Area of
Area of
Area of
Area of
•
1
= (L ∗ W) − πr 2
2
1 2
= (2r)(r) − πr
2
1
= 2r 2 − πr 2
2
1
= 4(2r 2 − πr 2 )
2
= 8r 2 − 2πr 2
Multiply by 4
Simplify
Area of a circle…
Area of a circle = πr 2
Multiply by 4 circles
𝐀𝐫𝐞𝐚 𝐨𝐟 𝐟𝐨𝐮𝐫 𝐜𝐢𝐫𝐜𝐥𝐞𝐬 = 𝟒𝛑𝐫 𝟐
Step 3. Carry out the Plan
•
By combining all these formulas altogether, we get…
𝐀𝐫𝐞𝐚 = (𝟒𝐫 𝟐 − 𝛑𝐫 𝟐) + (𝟖𝐫 𝟐 − 𝟐𝛑𝐫 𝟐) + 𝟒𝛑𝐫 𝟐
•
And then simplify
𝐀𝐫𝐞𝐚 = 𝟒𝐫 𝟐 + 𝟖𝐫 𝟐 + 𝟒𝛑𝐫 𝟐 − 𝟐𝝅𝒓𝟐 − 𝝅𝒓𝟐
𝐅𝐎𝐑𝐌𝐔𝐋𝐀 = 𝟏𝟐𝐫 𝟐 + 𝝅𝒓𝟐
Step 4. Check
Example 1:
Find the area covered by the elastic band.
1cm
Solution: let r = 1cm
𝐴𝑟𝑒𝑎 = 12r 2 + 𝜋𝑟 2
= 12(12 ) + 3.14(12 )
= 12 + 3.14
≈ 15.3.14 𝑐𝑚 2
∴ The area covered by the elastic band as it wraps four circles is approximately 15.3.14 𝑐𝑚 2 .
ANALYSIS AND RESULTS
The area of 15.3.14 𝑐𝑚 2 is quite reasonable since if we stretch the edges of the band
into a perfect square, the area would be simply (4)(4) = 16𝑐𝑚 2 . We can infer that the
area of the band inscribing four circles is lesser than the area of a square inscribing
four circles. It is because there is no space in the corners and there are no additional
gaps on the edges when an elastic band is wrapped compared to a square.
CONCLUSION
To conclude, finding the area covered by an elastic band around four circles is
equivalent to adding all the area of the shapes present inside the elastic band. By
applying the method in finding the area of composite shapes we were able to come up
with a simplified formula in getting the area covered by an elastic band given the
measurement of the radius.
This formula can be utilized in all forms of elastic band wrapping four circles
relevant to the figures and examples presented in this paper. It will give the same
result despite having different formations and arrangement of the circles inside the
elastic band as it shows the same number of circles and sections of the gaps.
FUTURE RESEARCH
Based on the results and discussions of this study, there is a need to investigate
the newly developed area formula determining the areas inside of elastic bands. The
validation of the proposed formula will be undertaken through experimentation.
Furthermore, the research aims to delve into generalizations, potentially extending the
formula to accommodate variations in the number of circles enclosed within the
bands.
ACKNOWLEDGEMENTS
This intricate tapestry of our mathematical investigation project wouldn't exist
without the invaluable contributions of several guiding threads. We, the researchers,
would like to extend our deepest gratitude to the following individuals and groups who
have played a significant role in this project.
To our instructor for M3112: Problem-Solving, Mathematical Investigation and
Modeling, Sir Mildin J. Retutas, for this enriching experience and opportunity to
engage in a Mathematical Investigatory Project under his profound guidance,
unwavering support, and expert mentorship.
To our supportive peers, though not directly involved in our research, their
willingness to share insights, offer clarification, and engage in discussions significantly
enriched our learning environment.
To our families and friends, their unwavering moral support and continual
encouragement were a constant source of motivation throughout this academic
journey.
Above all things, to our Almighty God. We are thankful for the boundless
fountain of knowledge and strength you have showered upon us. And, of course, for
sending the individuals mentioned above to guide us on this path.
APPENDICES
REFERENCES
Vocabulary.com. (n.d.). Elastic band. In Vocabulary.com Dictionary. Retrieved
January 01, 2024, from https://www.vocabulary.com/dictionary/elastic band
Merriam-Webster. (n.d.). Elastic band. In Merriam-Webster.com dictionary. Retrieved
January 2, 2024, from https://www.merriam-webster.com/dictionary/elastic%20band
Department of Materials Science and Metallurgy - University of Cambridge. (n.d.).
Theory of rubber conformation. Creative Commons Attribution - NonCommercialShareAlike
2.0
UK:
England
and
Wales
License.
https://www.doitpoms.ac.uk/tlplib/stiffness-of-rubber/rubber-conformation.php