angles of elevation of the pyramids of egypt

ANGLES OF ELEVATION OF THE PYRAMIDS OF EGYPTAuthor(s): ARTHUR F. SMITHSource: The Mathematics Teacher, Vol. 75, No. 2 (February 1982), pp. 124-127Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27962809 .

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ANGLES OF ELEVATION OF THE PYRAMIDS OF EGYPT

By ARTHUR F. SMITH Rhode Island College Providence, RI 02908

Only an unfortunately blas? person could stand at the foot of the Great Pyra mid of Giza without experiencing waves of wonder at its very existence. Every thing about it is astonishing?its beauty, its bulk, its history, its mystery. By day it is awesome?a man-made mountain that

originally towered 147 meters above its

base; a square, 230 meters on a side; and

covering some 5 lA hectares of desert. By night, a looming mass under a starry sky, it seems what its builders might very well have intended it to be?a stairway to heaven.

It is easy enough to measure the dimen sions of the pyramids of Egypt. But a

mystery arises when one contemplates the

angles of elevation used in their construc tion. (The angle of elevation is the angle between the edge of the base and the slant

height, the line from the apex of the pyramid to the midpoint of any side of the base. It is the maximum possible angle of

ascent for anyone attempting to climb to the top.) Were these angles arbitrary? If

not, how were they selected? These ques tions have never been definitively an

swered, and your theories, or those of

your students, may be as good as any one's. Some of the factors involved, how ever, must surely be associated with the evolution of pyramid building.

Throughout Egypt there are some

eighty pyramids. More are found in Su

dan, south of Egypt. Almost all of Egypt's pyramids are situated on the western edge of the Nile Valley, reaching south from Cairo for fifty miles. Yet, only a handful are of sufficient size and quality to cause men to marvel. Of these, the most promi nent are at Giza, a suburb of Cairo.

The Giza pyramids were constructed as monumental tombs. Three are major pyra mids, and the northernmost one is named for its pharoah-builder, Khufu (better known to us by his Greek name, Cheops). Known as the Great Pyramid, it is the

largest ever built (fig. 1). The construction

required three decades of toil by thou sands of workers whose numbers were

Fig. 1. The Great Pyramid of Khufu (Cheops) at Giza

124 Mathematics Teacher



swelled by farmers for perhaps three months of each year when flooding of the

Nile forced a lull in agricultural activity. They cut, transported, and positioned some 2 300 000 blocks of limestone and

granite, each weighing an average of about 2300 kilograms. At the time of the construction (there is considerable dis

agreement about dates: the National Geo

graphic Society puts it at about 2570 b.c., when Khufu's reign began), the pulley and the wheeled vehicle were not known to

the Egyptians, although they did use the lever and could move heavy blocks by

rolling them on logs. The fact and the

accuracy of the construction represent mathematical and mechanical achieve ments of the first magnitude.

The second, and middle, pyramid of the three major pyramids at Giza was built by

Khafre (Chephren), son of Khufu. It is

only slightly smaller in all dimensions than the Great Pyramid and has as a compan ion the famous Sphinx, whose face is said to be that of Khafre (fig. 2). Southernmost and last constructed of the Giza trio is the

pyramid of Menkaure (Mycerinus). It is much smaller than its neighbors, with a

height of about 66 meters.

Before the construction of the pyra mids, most tombs of Egyptian kings and

nobles were mastabas, low buildings of

rectangular shape. The oldest structure to be identified as a pyramid rather than a

mastaba is the Step Pyramid, built at

Saqqara by King Zoser about a century earlier than the Great Pyramid (fig. 3). An extension of the mastaba shape, this pyra mid looks like a six-layer cake, each layer (mastaba) smaller than the one below it.

Attaining a height of 60 meters, it has a

rectangular base, 125 x 109 meters. To the south of Saqqara some years later,

King Snefru ordered the construction of a "true" pyramid?one without steps? with an angle of elevation of 54Vi?. Some scholars theorize that when the structure had risen to a height of 49 meters, word was received of the collapse of part of another pyramid that had been under con struction at an angle of 52?. Snefru's archi tects altered the angle of his pyramid to

431/2?, and the structure, now known as the Bent Pyramid, was completed to a

height of 101 meters. Snefru later built the Red Pyramid near

by, entirely at the "safe" angle of 431/2?. It is considered the first successful "true"

pyramid. In order to achieve its height of 99 meters at the comparatively small angle of elevation, its base had to be immense

(220 square meters) and much more mate

February 1982 125



Fig. 3. The Step Pyramid at Saqqara

rial and human effort had to be expended than if the angle had been greater.

The technology acquired in these ex

periments enabled the Egyptians to build true pyramids at the steeper angle of 52?

(at Giza) and thus was ushered in the so called Age of the Pyramids, which lasted about a century. Later pyramids were much smaller; or of a different type; or

shoddy in design, material, or execution. Let us now consider the thematic ques

tion,

Why did the Egyptians build pyramids using angles of elevation of approxi mately 43V2? or 52??

In experimenting with the measure ments of Khufu's pyramid, one researcher divided the length of a side of the base by the height:

231 meters 3.14 . ?-? 1.57 ? ??

147 meters 2

Was this coincidence, or did the Egyp tians have this fine an approximation for

pi a thousand years before the time of the Ahmes (or Rhind) Papyrus? After all, the Ahmes Papyrus (ca. 1700 b.c.) represents the oldest firm evidence that the Egyp tians knew a value for , which was then

approximated as 3.16. One suggestion (Mendelssohn 1974) is

that the Egyptians might have employed

pi without realizing it. They might have measured long horizontal distances by means of a circular drum, or trundle, with some convenient diameter such as one cubit (based on the distance from elbow to middle fingertip). Such a trundle's circum ference would have been pi cubits. Per

haps, in order to design a pyramid of convenient and attractive proportions, the

Egyptians used a 1:4 ratio: for one revolu tion of the trundle along the horizontal line from the center of the base of the

pyramid to the midpoint of any side of the base, the height would rise four cubits. The ratio of the pyramid's side to its height would then be 2 :4, or : 2, the ratio actually encountered in the Great

Pyramid (see fig. 4). The angle of eleva tion of the slant height would consequent

Fig. 4

126 Mathematics Teacher



ly be arctan (41 ) ~

52?, as indeed it is for the Great Pyramid and several others.

To achieve a smaller angle of elevation, as in the case of the Red Pyramid and the

upper portion of the Bent Pyramid, a 1:3 ratio might have been used. In that case, the angle would be arctan (3/7 )

~ 43 Vi?. The use of 1:2 or 1:5 ratios would have resulted in designs with slopes too small to be attractive or too great to be feasible of construction.

If this theory is correct, the mechanical

accuracy involved in constructing the

trundle, and using it without slippage, would indeed be remarkable.

An individual climbing Khufu's pyra mid, as many visitors do, would probably climb not via the slant height route of 52? but via one of the ridges, from a corner of the base to the summit. The angle of ascent in that case would be arctan (147/ 115 /2)

? 42?, a much safer endeavor.

Today we can only theorize about why

the Egyptians used particular angles of elevation. Each year additional artifacts and hieroglyphics are discovered. Perhaps new clues will provide a clear solution to this mystery. Meanwhile, we will contin ue to measure the pyramids and to wonder at them.

BIBLIOGRAPHY

Billard, Jules ., ed. Ancient Egypt. Washington, D.C.: National Geographic Society, 1978.

Casson, Lionel, ed. Ancient Egypt. New York: Time-Life Books, 1965.

Edwards, I. E. S. The Pyramids of Egypt. New York: Penguin Books, 1961.

Fakhry, Ahmed. The Pyramids (2d ed.). Chicago: University of Chicago Press, 1974.

Mendelssohn, Kurt. The Riddle of the Pyramids. New York: Praeger Publications, 1974.

I would like to acknowledge the assistance of

Edwin Rosenberg, Wesleyan University, Middle

town, Connecticut, in the preparation of this manu

script.

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February 1982 127



angles of elevation of the pyramids of egypt

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