# How Egyptian Pyramids were Built, Their Geometry and Proportions

#### Introduction

Pyramids were the initial type of structures and housing in the ancient, which individuals become attached to and loved them for many centuries. They are one of the oldest civilizations in the global history concerning construction and structures. The pyramids of Egypt were among the typical pyramids in the very old period, which numerous individuals were familiar with and appreciated. Moreover, they were constructed of abundant monuments. The pyramid’s bases were square, and the floors were very smooth (Bard pg11). On the outside, they assumed a triangular shape. From the square base, the pyramid walls rose to form a pointed top. Most of them were planned like the pharaoh’s tomb. It is thought that the embracing of this idea had been impacted by the Egyptians’ religion. For example, the Egyptians believed that Ra, the god of all the pharaohs, formed himself from a mold of earth that resembled the pyramids, and after that, he formed other Egyptian gods. Moreover, the Egyptians also deemed that the pyramids signified the sun’s rays. The utilization of these pyramids began in about 2700BC in Egypt whereas the ones that were predestined for the royal families started many thousands of years later and continued until 1700BC. When the 3rd generation of the Pharaohs was in reign is when the initial pyramid was constructed in Egypt (Bard pg13). King Djoser built this first pyramid. The structural design that was utilized to construct pyramids during this period was referred to as Imhotep. It was employed in building pyramids by stacking the mastabas (rectangular shaped second-hand supplies since the pharaohs usually buried them during their control. Making a step to the next stage or level required a sum of six mastabas. Some of the earliest pyramids in Egypt recognized by historians that were devised for the kings of Egypt are the pyramids of Giza. The paper aims to discuss how these Egyptian pyramids were built, their geometry and proportions, and the reasons as to why they were aligned very perfectly imitating Orion’s Belt.

#### Construction of the pyramids

Many years have passed, and numerous modern plans have come up, which have changed structural designs or architectures. In spite of such changes, many people have hypothesized how the magnificent pyramids were constructed. Different individuals adopt varying beliefs on the ways the pyramids were made. Some of them suppose that these structures were constructed by an extraterrestrial. Conversely, some individuals suggest that that was the knowledge that was used by the Egyptians but was lost as the time went on (Leeuwen pg163). Though numerous individual considers that the building of these pyramids was complicated, it was not the case in that period. For instance, it is estimated that about 25000 workers were involved the construction of the well-known pyramid of Giza for almost twenty-three years in comparison to the building of the Notre Dame Cathedral, which took about 200 years to be completed. Many of the present Egyptian pyramids were constructed during the reign of the pharaohs. Immediately a pharaoh ascended to control; he started constructing his pyramids. A committee that comprised of chief engineers and various architects was formed to oversee the building project. Their responsibility was to formulate the designs and to make sure that they were completed within the shortest time. A close observation of the existing pyramid shows that they were built on the western side of the Nile River, and this was to make sure that the sun’s rays at sunset could meet with the Pharaoh’s soul. Hence, two significant factors were considered when selecting the pyramid-building site. One was the nearness to the major city of ancient Egypt, Memphis and the direction of the sunset (Bard pg14).

Limestone formed the basis of the pyramid. It was a coarse limestone and was responsible for making a strong foundation. However, the inner layers were built utilizing better quality and finer limestone. This idea was responsible for making the interior wall look attractive and stunning with a white gleam. The inner walls were so shiny in that they could be noticed from afar. Moreover, they were also built utilizing capstones. The composition of the capstones comprised of granite, basalt or was a very tough stone that was gold or silver-plated. Plating was accountable for the reflective characteristic more so when there was a bright sun. Moreover, the pyramid got their magnificent look from the same (Margaret pg33).

Contrasting what many people think, the pyramids were not built by the slaves but the source of labor was from the peasants. In this period, slavery had not cropped up, and the labor could be tapped from the peasants. The laborers who were involved in the building of the pyramids were exempted from particular taxes. Besides, they were offered with primary needs that include food and clothing. Finally, they were provided with the good working environment during the construction of these structures.

Quarries were very useful since the building materials were tapped from them. Soft materials such as sandstones and limestone were extracted utilizing the copper chisels while harder stones needed the application of stronger and harder tools. Huge dolerite that made the granites blocks softer were used in the excavation of stones from quarries. About sixty and seventy men were involved during the pounding out of the stone. After that, pieces of wood were crammed underneath it in the open joints make the breaking of the stone easier (Túnyi et al pg30). These openings were subsequently filled with water, which softened the stones as well as increasing the speed of breaking. After full extraction of the stones, they were transported to the construction site using boats. Upon the arrival on the land, the stones were slipped on a greased slipway to the site (Margaret pg 40). Ramps were utilized to lift and fit the blocks in the pyramid upon reaching the construction site. The access ramps were made to make the lifting of the construction stones to the pyramid’s upper course easier. These ramps were made of mud bricks and were harden on the surface by pieces of plaster. Moreover, they could be shifted from one position to another as the structures moved one course up.

*The geometry and proportions of Pyramid at Giza*

*The geometry and proportions of Pyramid at Giza*

In the geometry and the measurement evaluation of antique structures, it is normally significant to merge mathematical and scientific neutrality with open minds. These will allow knowledge of what seems to be the goal of the constructor. Therefore, researching texts on the analyzed pieces of evidence and dependable researchers that have done the same studies before will help in the collection of such information. The combination of this method with photographs, images, and drawings with measurement, a reasonable research, can be agreed to. Researchers have obtained research and arithmetical data to hold their ideas about the Egyptian pyramids. The discussion will focus on the largest pyramid that was built at Giza specially made by Pharaoh Cheops or the Great Pyramid of Khufu (Bonwick pg55). The information of the pyramid in this discussion includes mathematical formulae such as pi, squaring of the globe, and the size of the earth in addition to other esoteric and numerical data. The first focus is on the confirmation of the height and base of measurement of the pyramid of Khufu at Gizeh in Egypt and the accuracy regarding Pi, Phi, and Pythagorean Theorem. The value, 22/7 is the employed for the Pi. The list below shows the dimensions of the great pyramid of Giza.

*Golden Ratio proportions as shown by the great pyramid*

*Golden Ratio proportions as shown by the great pyramid*

There has been a debate concerning the geometry utilized in the building of the pyramid of Khufu that was constructed around 2560BC. The smooth and flat outer wall walls have eroded over time and what has remained is the coarsely shaped internal core. Therefore, it becomes hard to determine the geometry with complete sureness. At the core, the outer shell remains to make it possible for the establishment of the original proportions. Evidence exists, showing that the pyramid design may have followed some of the mathematical and geometry foundations such as pi, phi, and the Pythagorean Theorem. Hence, there are several theories available to explore these three concepts (Butzer et al pg3352).

Phi based pyramid varies by 0.025% from the Khufu’s pyramid of Giza

Phi is the only figure that has the arithmetical characteristic of its square being one more than it is and can be represented as

1.618… + 1 = 2.618 or Φ + 1 = Φ²

When the Pythagorean equation is applied to this one can construct a right-angled triangle of the sides √Φ, 1 and Φ, as shown below and hence results in a pyramid with a base of 2 and height of 1.272 since it is the square root of phi. Therefore, the ratio of the height to the base is 0.636 (Taylor pg632).

In agreement with findings from several studies, this pyramid has a base of 755.9 feet (230.4 meters) and an approximated height of 480.6 feet (146.5 meters). This results in the height to base quotient of 0.636 that shows it is a Golden Triangle as rounded off to three significant decimal places. In the instance that the base measures exactly 230.4 meters, then the ideal golden ratio would need a height of 146.5367 meters. Hence, the variation is from the approximated dimensions of the pyramid is only by 0.0367 meters, which is 0.025%.

Based on Pi, a pyramid differs by only 0.1% from the Khufu’s Pyramid projected measurements

Moreover, by constructing a circle comprised of a circumference of 8 units, which is similar to the perimeter of the pyramid made of a base of 2 units then folding the arc of the semi-circle, the height with being the radius. Hence, 8/pi/2 or 1.273 gives the radius. It is only 1/10th different from the height, 1.272 of the golden triangle. Using the height (146.5 meters) of the Great Pyramid would give a difference of 0.14m that is approximately 0.1% difference from the golden triangle (Butzer et al pg3356).

Based on a regular gradient a pyramid differs only by 0.8% from the Khufu’s Pyramid estimated measurements

Moreover, there is another probability that the great pyramid of Giza I based on a method referred to as Seked. It is a measure of the gradient. It was based on the Egyptian measurement system where one cubit equals seven palms while the palm equals four digits (Werner pg100). The theory suggests that the Khufu’s pyramid applies the gradient of 5.5 sekeds. Hence, a pyramid of one cubit is supposed to have a base of 5.5 palms. Hence, the ratio of the height to the base id 7/5.5 resulting in 1.2727. Therefore, it is closely related to the golden triangle and the square root of phi. The pyramid constructed employing Phi would give a gradient of 51.83 while that created using sekeds would result in 51. 84. If the Khufu’s pyramid were created using seed, the height would have been 146.618 meters and the same base of 230.4 meters and hence a difference in height by 0.118meters from the actual estimated height and hence giving a variation of 0.8% from the golden triangle base on pi or phi (Spence pg321).

The pyramids of Giza can be seen as an ideal reflection of the three stars of Orion’s belt. Just like these pyramids, the stars are not ideally aligned since the smallest one offsets slightly to the east. Moreover, all the three pyramids just like the stars are southwesterly slanted. The orientation of the three pyramids of the River Nile reflects Orion’s course to the Milky Way (Farhat pg754). Finally, the arrangement of the three pyramids and their comparative sizes were premeditated design plan resembling the Orion Belt. Therefore, the construction of the three pyramids of Giza discloses the “The Orion Mystery.” Hence, the pyramids were an imitation of heaven on earth. These pyramids were constructed with great accuracy to act as the pharaoh’s path to the heavens or the stars (Schoch pg243).

The shape of various asterisms and constellations adjusts over time because the stars are not stationary and move throughout the space. However, the stars that are found in Orion’s belt have the same appropriate movement as well as sharing the same origin. Hence, they travel in togetherness maintaining similar shape for many centuries. Therefore, Orion’s belt resembled what is seen today. The bright star Sirius and Orion symbolize Isis and Osiris according to the ancient Egyptian legends. They gods were believed to have descended from Orion’s belt and Sirius and started the human race. Therefore, the three pyramids replicate the arrangement of the three stars in the belt (Glaeser pg129). Moreover, the airshafts in the pyramids point straight in the direction of Orion constellation. Therefore, the ancient Egyptians believed that they were responsible for protecting the pharaoh’s soul to the Orion. The correspondence between the pyramids and the three stars is significantly extraordinary although this is just theory.

#### Conclusion

In summary, the building of these Egyptian pyramids in the ancient times was a very overwhelming experience for the people who were there to witness. Although the structural designs of that period were not as advanced as modern ones, they played a significant responsibility in inspiring the modern architecture to its present day position. It is very clear that the construction of the pyramids in the ancient Egypt was a highly organized process that took precision and construction knowledge. The building materials used were massive and heavy, and hence the skills such as the use of ramps and lubricated pathways and workforce to lift and fit them in the pyramid were needed. Moreover, the Egyptians went further to have knowledge of mathematics and geometry to ensure that the structures were accurate and durable. Therefore, it is lucid that the proportions and geometries that were used in the ancient Egypt did not occur by chance. This can be supported by the evidence that they were capable of aligning the pyramids of Giza to true north to within an accuracy of 1/15 of a degree. Hence, the Egyptians have the knowledge of geometries and proportions, and this helped them, mainly in the building of the Great Pyramid in addition to the median pyramid, which was a step pyramid with three tiers. Their ability to achieve the precise angle of 51.83 of the golden triangle show that they were very aware of what they were doing and hence they can be categorized as architects that were knowledgeable in mathematical geometry and proportions.

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