Mathematical Exploration

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INTERNATIONAL BACCALAUREATE
IB MATHEMATICS HL
INTERNAL ASSESSMENT
PYHTAGOREAN THEOREM IN EGYPTIAN PYRAMIDS
Serra Koz
IB Candidate Number: ---
May 2014

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INTRODUCTION
The pyramids that are one of the Seven Wonders of the World are symbols of Egypt and also
have a remarkable place in geometry. They have always fascinated the historians, the
mathematicians and the archeologists for centuries as Pythagoras was one of them. In the
structure of pyramids, Pythagorean’ theorem which is one of the major theorems of geometry
is mostly used. There are lots of methods of proving Pythagorean Theorem that use different
techniques of mathematics.
Within exploration, the reader will learn some methods of proofs of Pythagorean Theorem
and the usage of it in Egyptian pyramids. In addition, over the years, some changes occurred
in the dimensions of pyramids with the effects of natural disasters such as erosion. These
changes also made some differences in the volume of pyramids and this difference is going to
be calculated by using Pythagorean’s theorem.
PYTHAGOREAN THEOREM AND 3 WAYS OF PROOVING IT
The rule of the Pythagorean Theorem can be summarized as the sum of the squares of the legs
of a right triangle is equal to the square of the hypotenuse.
According to the right triangle given in Figure 1,
2 2 2
c =a +b
Figure 1

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As it is indicated in introduction, there are lots of proofs of the Pythagorean Theorem
however this mathematical exploration includes only 3 of them.
Algebraic Proof (Indian Proof)
Figure 2
In Figure 2, ABCD is a square with the length of a side (a+b). Area of ABCD is;
A(ABCD)=(a+b)(a+b)
The sides of ABCD is divided into two parts that their lengths are a and b respectively. Thus,
the inside figure shaded with pink becomes a square, too and the area of EFGH is;
A(EFGH)=c2
[Equation 1]
There are four small right triangles; each one has an area of;
1
A( EAF)=A( FBG)=A(GCH)=A( HDE)= ab
2
   
If the sum of the areas of the small triangles is A1; then
1
1
A =4( ab)=2ab
2
[Equation 2]
D
A B
C
E
F
G
H

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Thus, total area of the square ABCD is equal to the sum of the Equation 1 and Equation 2
which is
A(ABCD)=c2
+2ab
As the area of ABCD is indicated at the beginning of the proof as (a+b)(a+b), then
(a+b)(a+b)= c2
+2ab [Equation 3]
If the Equation 3 is rearranged;
a2
+2ab+b2
= c2
+2ab
a2
+b2
=c2
which proves the theorem.
Triangular Proof
Figure 3
All triangles in Figure 3 are right triangles with the lengths of the legs a and b and the length
of the hypotenuse is c. In this case, the area of each triangle is;
ab
Area=
2
In Figure 4, the triangles in Figure 3 are reorganized in order
to form a square with the length of a side is c;
Figure 4

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Inside the big square, there is a square hole with the side (a-b). The total area of the square
hole and 4 identical right triangles is;
(a-b)2
+2ab
As the area of the big square is c2
,
(a-b)2
+2ab= c2
[Equation 4]
After rearranging Equation 4;
a2
-2ab+b2
+2ab=c2
a2
+b2
=c2
Proof by Using Trapezoid
The trapezoid given in Figure 5 is a right trapezoid and according to
way to find the area of a trapezoid which is the half of the product of
height and the sum of the parallel sides,
A(ABCD)=
1
(a+b)(a+b)
2
=
2 21
(a +2ab+b )
2
Figure 5
Inside the right trapezoid, there are three right triangles and their areas are labeled as
A, B and C. Thus,
21 1 1
A= ab B= ab C= c
2 2 2

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As the sum of the areas of triangles will give the area of the trapezoid ABCD,
2
2 2 2
2 2 2
1
A+B+C= (ab+ab+c )=A(ABCD)
2
1 1
= (2ab+c )= (a +2ab+b )
2 2
c =a +b
PYRAMIDS
Of all the marvels of Ancient Egypt, the pyramids leave the most lasting and most
vivid impression. They not only affect the intellect but also arouse deep and uncontrollable
emotions. The old statistics which have been carefully complied by Egyptologists regarding
heights and volumes, number of blocks and their average weight, years of labor and the like,
leave the visitors only mildly impressed. (Ventura, 42)
Egyptians mostly used right triangles with the length of the
legs 3 and 4 and the hypotenuse is 5 in the pyramids and all the
pyramids that placed in Egypt consist of square pyramids
which have square bottoms. Square pyramids have 5 faces that
4 of them are triangular shaped, 8 edges and a square bottom.
USAGE OF PYHTAGOREAN THEOREM IN PYRAMIDS
This exploration includes six areas of usage of the Pythagorean Theorem and
operations applied according to the data of the Khufu Pyramid. Khufu Pyramid is one of the
Seven Wonders of the World and it is the only one exists today. The length of one side of the
base of the Khufu Pyramid is 230,40 meters. The original height of the pyramid is 146,49
meters but some natural disasters, such as erosion, affected the height of the pyramid. Today,
the current height of the Khufu Pyramid is 138,75 meters.
Figure 6

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As the height of the Khufu Pyramid and the length of one side of
the base are given, Pythagorean Theorem can be applied in order
to
find the slant length of the pyramid.
According to Figure 8,
230,40
b= =115,2 m
2
   
2 2 2
2
2
146,49 + 115,2 =c
21459,3201+13271,04=c
34730,3601=c
c 186,36 m



 
In order to calculate the edge of the triangular surface, 2 different ways can be shown.
1st
Way: According to the information below, the triangle of the surface is an isosceles
triangle, thus Pythagorean Theorem can be used to find the edge of the pyramid.
As the length of the a and b are prevously found, to find the value of c in Figure 7:
   
2 2 2
2
2
186,36 + 115,2 =c
34730,0496+13271,04=c
48001,0896=c
c 219,09 m


 
Figure 8
Figure 7

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2nd
Way: In the second way, diagonals have to be found. The
base of the pyramids is square, thus Pythagorean Theorem
can be applied in the green area shown in the Figure 9.
   
2 2 2
2
2
230,4 + 230,4 =x
53084,16+53084,16=x
106168,32=x
x 325,83 m


 
As x is the diagonal of the base, in order to apply the Pythagorean
Theorem to the purple region in Figure 10, the value of x should be
divided into 2.
x 325,83
b= = 162,92 m
2 2

Finally, by applying the theorem
   
2 2 2
2
2
162,92 + 146,49 =c
26542,9264+21459,3201=c
48002,2465=c
c 219,09 m


 
Figure 9
Figure 10

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After showing how to use Pythagorean Theorem in order to find some basic elements
of the pyramids, now the changes occur in the Khufu pyramid, specifically in its volume, will
be calculated.
As the Khufu Pyramid is a square pyramid, volume is calculated by the formula
21
V= b h
3
If the volume formula is applied to the dimensions of the Khufu Pyramid,
2
3
1
V= (230,4) .(146,49)
3
1
V= (53084,16)(146,49)
3
1
V .(7776298,598)
3
V 2592099,533 m


The volume of the Khufu Pyramid is calculated,
however as it is mentioned, there are some changes
occur in the dimensions of the pyramid such as its
height decreases from 146,49 m to 138,75 m shown
in Figure 11. Amount of change in the length of the
height is 7,74 m and it also affects the length of the
slant and the value of the volume.
Figure 11

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Now, by the following calculations, the current length of the slant and the current volume will
be found.
   
2 2 2
2
2
138,75 + 115,2 =c
19251,5625+13271,04=c
32522,6025=c
c 180,34 m


 
Thus, before the erosion the length of the slant was approximately 186,36 m and the effect of
the erosion decreases it to approximately 180,34 m. The amount of decrease is about 6,02 m.
By using the volume formula, the current value of the volume of the pyramid is found as
2
3
1
V= (230,4) .(138,75)
3
1
V= (53084,16).(138,75)
3
1
V= .7365427,2
3
V 2455142,4 m
The losses due to the Erosion can be found using the volumes of the Khufu Pyramid. The
difference between initial volume and current volume shows the effect of the erosion.
initial current
3
ΔV=V -V
2592099,533-2455142,4
136957,133 m



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CONCLUSION
This exploration investigates the usage of the Pythagorean Theorem in Khufu Pyramid and
shows how to use one of the basic theorems of geometry in real life. It is also important for
the international perspective because it proves the effects of natural disasters on the Khufu
Pyramid with the mathematical operations. There can be some other factors that change the
dimensions of the pyramids and by considering them, the future changes can be calculated
again for further explorations. By the help of these calculations, pyramids which are one of
the seven wonders can be protected. The data used in the exploration is approximately same
for the real measurements in the Khufu Pyramid.
REFLECTION
This topic is my area of interest as I am interested in architecture and examining the changes
happening in architectural works can be useful in protecting them from the external factors.

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REFERENCES/BIBLIOGRAPHY
i. Alawneh, Ameen and Kamel Al-Khaled. Pythagorean Theorem: Proof and
Applications. Jordan: Department of Mathematics and Statistics, Jordan University of
Science and Technology.
ii. Anonymous. “A Brief History of the Pythagorean Theorem” (n.d.) 27 October 2013.
Retrieved from: <http://www.geom.uiuc.edu/~demo5337/Group3/hist.html>
iii. Anonymous. “Pythagorean Theorem.” (n.d.) 27 October 2013. Retrieved from:
<http://www.cut-the-knot.org/pythagoras/>
iv. Anonymous. “Mathematics 1010 online: The Pythagoeran Theorem” (n.d.) 27
October 2013. Retrieved from: <http://www.math.utah.edu/online/1010/pyth/>
v. Carpicepi, Alberto Carlo & Giovanna Magi. Art and History Egypt. Translated by
Erika Pauli, Florance-Italy, ISBN: 978-88-476-2415-3
vi. Deb Russel, “Surface Area and Volume: Cylinderi Cone, Pyramid, Sphere” (n. d.) 28
October 2013 pg. 5. Retrieved from:
<http://math.about.com/od/formulas/ss/surfaceareavol_5.htm>
vii. Farrell, Sean. “The Great Pyrmid of Giza” (n. d.) 4 December 2013. Retrieved from:
<http://www.personal.psu.edu/spf5043/art002/Assignment7.htm>
viii. Ventura, Dr. R. Egypt History & Civilization. Chief Photographer by Garo
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(202)3911489, ISBN: 978-977-496-008-2
ix. Zijlma, Anouk. “Egypt- Egypt Facts and Information” (n. d.) 28 October 2013.
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