Short-cut Formulas in Finding the Areas of a Shaded Region Between the Circle and a Square of an Inscribed Circle in a Square and Circumscribing Circle to a Square

1. Short-cut Formulas in Finding the Areas of a
Shaded Region Between the Circle and a Square
of an Inscribed Circle in a Square and
Circumscribing Circle to a Square
An Investigatory Paper Presented To:
The Faculty of Tandag National Science High School
Myrvic O. Laorden
S.Y. 2015-2016

2. ABSTRACT
The investigator came up with the study because she wanted to help the students in
solving the area of shaded region of an inscribed circle in a square and area of the circumscribing
circle to a square in an easier and faster way. She wanted to widen the knowledge about the
relationship of the two plane figures whether it is inscribed or circumscribed.
In doing such project, she really tried her best to have accurate solutions. She gathered
many reference books as she could and got some information from the internet.
The investigator found out that the new found short-cut formula was derived because
there is a constant relation to the area of the square and the inscribed circle. With the help of the
constant value ( ) that has been calculated, we can easily get the area of the inscribed
circle in a square with the given measurement of the edge; even without the help of the radius of
the circle, through multiplying the constant value ( ) into the area of the square. The new
formula is:
Area of the shaded region = (area of a square) (constant)
In symbols: Ashaded region = ( )
Likewise on the circumscribing circle to a square the new found short-cut formula in
finding the shaded area was also derived. With the help of the constant value ( ) that has
been calculated, we can easily get the area of the circumscribed circle to a square with the given
measurement of the edge; even without the help of the radius of the circle, through multiplying
the constant value ( ) into the area of the square. The new formula is:
Area of the shaded region = (area of a square) (constant)
In symbols: Ashaded region = ( )
The researcher found out that we can now get the area of the shaded region using only the
sides of square with the constant values. So getting away from the tedious long process of
finding the areas of the two figures and their difference.

3. TABLE OF CONTENTS
ABSTRACT
ACKNOWLEDGEMENT
Chapter I
Background of the Study
Objectives of the study
Significance of the Study
Scopes and Limitations
Review of Related Literature
Chapter II
Materials
General Procedure
Safe – keeping device
Testing the devise
Chapter III
Results and Discussions
Chapter IV
Conclusions and Recommendation
REFERENCES
CURRICULUM VITAE

4. ACKNOWLEDGEMENT
The investigator expresses her sincere thanks and heartfelt appreciation to the following
individuals whose invaluable assistance and cooperation made the completion of the study
possible.
To Mr. and Mrs. Romulo T. Laorden the beloved parents as well as the teachers for
guiding her during the investigation and sharing their knowledge not only in mathematics but in
general knowledge that gave them great ideas.
To the rest of the faculty members and students of Tandag National Science High School
for the extra help and understanding.
And above all, God Almighty for the strength and wisdom He had given them to
accomplish this study.

5. Chapter I
Introduction
Background of the Study
In most complex problem in Geometry, finding the area of shaded region of an inscribed
circle in a square and a circumscribed square in a circle are often given. Solving the area of the
shaded region in an inscribed circle is quite time-consuming. We need to get first the area of a
square, and the area of a circle using their formulas which is: and
respectively. After getting the area of square and circle, we can now find
the area of shaded region of an inscribed circle in a square by subtracting the area of square
minus area of a circle, or .
Moreover, to get the area of the shaded region of a circumscribing circle, when given an edge of
a square, the main astray to this problem is that we cannot directly get the area of a circle
because we need to find first the diameter of the circle by using the Pythagorean Theorem since
upon doing such, we only get the length of the hypotenuse of a square which is congruent to
diameter of a circle. After getting the diagonal of a square, which is also the diameter of a circle,
we can now start solving the areas of the two plane figure using again the formula written above
and of the same process in finding the area of the shaded region of a circumscribing circle to a
square which is .
Answering these kind of problems especially in mathematical competitions need to be
exact but consumes less time in order to win. In coherence, the researchers wanted to help math
wizards to achieve these requirements in order to attain an accurate answer in a short span of

6. time, by finding a short-cut formula in finding the shaded region of to plane figure when
inscribed or circumscribed.
Objectives of the Study
1. To determine the constant value that can be used in making an easier equation to find the
area of the shaded region between the circle and a square of a circle inscribed in a square
and for that circle circumscribes a square.
2. To derive short-cut formula in finding the area of shaded region between the circle and a
square of an inscribed circle in a square.
3. To derive short-cut formula in finding the area of the shaded region between the circle
and a square in a circumscribing circle to a square.
Significance of the Study
The significance of this study is that it can add new knowledge in the field of
Geometry, specifically about deriving short-cut formula in finding the area of shaded
region of an inscribed circle in a square and area of the shaded region of a circumscribing
circle to a square. It can give the students an easier way to find the area, even if given
only the length of side of a square which it circumscribe or in which it is inscribed. It
would be of great help to the students, especially those who join in math contests.
Scopes and Limitations
This study is only limited to the use of short-cut formula in finding the area of shaded
region of an inscribed circle in a square and area of the shaded region of a circumscribing circle
to a square.

7. Review of Related Literature
Circle
The word "circle" derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself
a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring". The origins of the
words "circus" and "circuit" are closely related. E Circle was known before the beginning of the
history records and have been observe in the images like Moon, Sun, and a short plant stalk
blowing in the wind on sand, which forms a circle shape in the sand. The inventors of wheel,
which, with related inventions such as gears that makes much of modern machinery possible got
an idea in circle. In mathematics, the study of the circle has helped inspire the development of
geometry, astronomy, and calculus.
In mathematics, a careful distinction is made between the parts of circle, its primary parts
(circumference, arcs and center) and its secondary parts (radii, diameter, chord, secant and
tangent)
Area of the Circle
The distance around a circle is called its circumference. The distance across a circle
through its center is called its diameter. We use the Greek letter (pronounced Pi) to represent
the ratio of the circumference of a circle to the diameter. In the last lesson, we learned that the
formula for circumference of a circle is: . For simplicity, we use = 3.14. We know

8. from the last lesson that the diameter of a circle is twice as long as the radius. This relationship is
expressed in the following formula: .
The area of a circle is the number of square units inside that circle. If each square in the
circle to the left has an area of 1 cm2
, you could count the total number of squares to get the area
of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26
cm2
However, it is easier to use one of the following formulas:
or
where is the area, and is the radius and = 3.14 in our calculations.
Inscribed circle
An inscribed circle is the largest possible circle that can be drawn on the inside of a plane
figure. For a polygon, each side of the polygon must be tangent to the circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits
snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G"
means precisely the same thing as "figure G is circumscribed about figure F". A polygon
inscribed in a circle, ellipse, or polygon has each vertex on the outer figure. Common examples
of inscribed figures include circles inscribed in triangles or regular polygons, and triangles or
regular polygons inscribed in circles.

9. Circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle which
passes through all the vertices of the polygon. The center of this circle is called
the circumcenter and its radius is called the circumradius.
A polygon which has a circumscribed circle is called a cyclic polygon (sometimes a concyclic
polygon, because the vertices are concyclic). All regular simple polygons, isosceles trapezoids,
all triangles and all rectangles are cyclic.
A related notion is the one of a minimum bounding circle, which is the smallest circle
that completely contains the polygon within it. Not every polygon has a circumscribed circle, as
the vertices of a polygon do not need to all lie on a circle, but every polygon has a unique
minimum bounding circle, which may be constructed by a linear time algorithm. Even if a
polygon has a circumscribed circle, it may not coincide with its minimum bounding circle; for
example, for an obtuse triangle, the minimum bounding circle has the longest side as diameter
and does not pass through the opposite vertex.

10. Square
A Square is a flat shape with 4 equal sides and every angle is a right angle (90°). It is also
fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length). It
has properties of:
 All sides are equal in length
 Each internal angle is 90°
 Opposite sides are parallel (so it is a Parallelogram)
 Diagonals are equal
Area of a Square
If the sides of a square have length s, the perimeter of the square is simply four times the
length of a side, represented algebraically by the formula P = 4s. The area of a square is
determined by multiplying the length of a side by itself:
Area = s2
= s × s
Units
Remember that the length of an edge and the area will be in similar units. So if the edge length is
in centimeter(cm), then the area will be in square centimeter(cm
2
) and so on.

11. Chapter II
Methodology
Materials
 Drawing materials
 Ruler
 Paper and pencil
 Calculator
 Compass
General Procedure
Finding the short-cut formula of the area of shaded region between the circle and a square of
inscribed circle in a square
The investigators assigned different lengths of the edge of the square from 1cm-10cm.
They then solve for the area of the shaded region of inscribed circle in a square using the existing
formula, A= for square and A= for circle. Further, they compared the formula of finding
area of the two plane figures and look for their common. Then, they solve for the constant value
by rewriting the factors, putting common factors outside parenthesis. The investigators simplify
the values inside the group symbols to get the constant value, and formulated a new
formula/relation in finding the area of shaded region of inscribed circle in a square using the
measure of the edge of inscribing square. Refer to Table 1

12. Derivation for the short-cut formula in finding the shaded area between the square and a
circle of an inscribed circle in a square:
Let s be the length of an edge of a square:
Provided that the radius of the circle is equal to ½ of the edge of the square; r = s and
Substituting the value of radius using the side s:
( )
A = A = ( ) A = ( )
A = ( )
Derivation of new formula:
Area of the shaded region = (area of a square) (constant)
In symbols: Ashaded region = ( )
S
S

13. Finding the short-cut formula of the area of shaded region between the circle and a square of
circumscribed circle to a square
Likewise, the investigators assigned lengths of the side of the square from 1cm-10cm.
Next, they solve for the lengths of the radius using the Pythagorean Theorem. Then using the
original formula (A= and A= ), the area of shaded region of circumscribed circle to a
square are solved. They they compared the formula of finding area of the two plane figures and
look for their common. Then, they solve for the constant value by rewriting the factors, putting
common factors outside parenthesis. The investigators simplify the values inside the group
symbols to get the constant value. After such, they derived new formula/relation in finding the
area of shaded region of circumscribed circle to a square using the measure of the edge of
inscribing square.
Derivation for the short-cut formula in finding the shaded area between the circle and a
square of a circumscribed circle to a square:
Let r be the radius, s be the length of an edge of a square:
Provided that the radius of the circle is equal to ½ of the edge of the square; r = s and
Before we can solve the shaded area, we need first to solve the diameter of a square
which is equal to the diagonal of a square. Using the Pythagorean Theorem where
s
s

14. c is the hypotenuse, a and b are sides of a square. Since hypotenuse is equal to the diagonal and
diameter of a circle and a and b are also the sides of the square, we can solve using this pattern:
( )
Solving the shaded region using or r = , therefore
( )
A = ( ) A = ( ) A = ( )
A = ( )
Derivation of new formula:
Area of the shaded region = (area of a square) (constant)
In symbols: Ashaded region = ( )
Safe-keeping Device
Finding new thing is equivalent of putting new responsibility. That’s the reason why in
every devices you bought, it includes the passport with the terms and condition so that
consumers will be responsible.

15. In making this study the researchers are also very careful and give more time and efforts
to evaluate this study and determine how useful and effective it is in the part of the learners.
Testing the Device
Table 1 and two shows the affeciency of the new derived short-cut formula in finding the
area of shade region of an inscribed and circumscribed circle about a square.

16. Chapter III
Results and Discussion
Tabulated result of finding relation of Inscribed Circle in a Square given the length
of an edge of a Square
Table 1 (Inscribed circle in a square)
New found short-cut formula in finding the shaded area of inscribed circle in a square
Length
of the
Edge
Area of a
Square
Area of a
Circle
Shaded Area
using
Constant
Value to be
multiplied
to the Area
of Square
Shaded Area using
Derived Formula
( )
Remarks
1 cm 1 0.785 0.215 0.215 0.215
Exactly the
same
2 cm 4 3.148 0.860 0.215 0.860
Exactly the
same
3 cm 9 7.069 1.935 0.215 1.935
Exactly the
same
4 cm 16 12.566 3.440 0.215 3.440
Exactly the
same
5 cm 25 19.635 5.375 0.215 5.375
Exactly the
same
6 cm 36 28.274 7.740 0.215 7.740
Exactly the
same
7 cm 49 38.485 10.535 0.215 10.535
Exactly the
same
8 cm 64 50.265 13.760 0.215 13.760
Exactly the
same
9 cm 81 63.617 17.415 0.215 17.415
Exactly the
same
10 cm 100 78.540 21.500 0.215 21.500
Exactly the
same

17. Table 1 reveals that the constant value of 0.215 when multiplied to the area of the square
gave exactly the value using the subtraction method. The investigators verified the table and
found out that the new found short-cut formula in finding the area of the shaded region of an
inscribed circle in a square is equal to the area using the usual method.
Table 2(Circumscribing circle to a square)
New found short-cut formula in finding the shaded area of circumscribing circle to a square
Length
of the
Edge
Area of a
Square
Area of a
Circle
Shaded Area
using
Constrant
Value to be
multiplied to
the Area of
Squarea
Shaded Area using
Derived Formula
( )
Remarks
1 cm 1 0.1.57 0.570 0.57 0.570
Exactly the
same
2 cm 4 6.28 2.280 0.57 2.280
Exactly the
same
3 cm 9 14.13 5.130 0.57 5.130
Exactly the
same
4 cm 16 25.12 9.120 0.57 9.120
Exactly the
same
5 cm 25 39.25 14.250 0.57 14.250
Exactly the
same
6 cm 36 56.52 20.520 0.57 20.520
Exactly the
same
7 cm 49 76.93 27.930 0.57 27.930
Exactly the
same
8 cm 64 100.48 36.480 0.57 36.480
Exactly the
same
9 cm 81 127.17 46.170 0.57 46.170
Exactly the
same
10 cm 100 157 57.000 0.57 57.000
Exactly the
same

18. Table 2 reveals that the constant value of 0.57 when multiplied to the area of the square
gave exactly the value using the subtraction method. The investigators verified the table and
found out that the new found short-cut formula in finding the area of the shaded region of
circumscribed circle to a square is equal to the area using the usual method.

19. Chapter IV
Conclusion
 Inscribed circle in a square
The researchers therefore conclude that the new found short-cut formula was derived
because there is a constant relation to the area of the square and the Inscribed circle. With the
help of the constant value ( ) that has been calculated, we can easily get the area between
the circle and a square of the inscribed circle in a square with the given measurement of the edge;
even without the help of the radius of the circle, through multiplying the constant value
( ) into the area of the square.
 Circumscribing Circle to a Square
The researchers therefore conclude that the new found short-cut formula was derived
because there is a constant relation to the area of the square and the circumscribed circle. With
the help of the constant value ( ) that has been calculated, we can easily get the area
between the circle and a square of the circumscribed circle to a square with the given
measurement of the edge; even without the help of the radius of the circle, through multiplying
the constant value ( ) into the area of the square.
Recommendation
Based from the findings and conclusions of the research, further studies must be
conducted to validate the use of the new found formulas can be used in other plane figure that is
also inscribed and circumscribed.

20. REFERENCES
Microsoft Encarta Dictionary
Microsoft Encarta Premium DVD 2009
Internet Explorer
Calculator

21. CURRICULUM VITAE
PERSONAL BACKGROUND
Name: Myrvic O. Laorden
Nickname: Gaga
Age: 15 years old
Parents: Mr. Romulo T. Laorden
Mrs. Mia O. Laorden
Hobbies: Dancing, Singing, Playing the guitar, reading, watching tv, surfing
the internet
EDUCATIONAL BACKGROUND
Elementary: Special Science Elem. School
Tandag City
High School: Tandag National Science High School
Tandag City