The shapes of the pyramids are determined by the angle of elevation of the four sides of the pyramids. These angles vary from pyramid to pyramid but are typically between 40 and 55 degrees for the larger and more famous Egyptian pyramids.

There has been considerable conjecture over the years as to how the Egyptians decided upon the slopes of the pyramids. They did not use the measure of degrees to determine angles. They apparently measured slopes by the ratio of the rise over run (change in height divided by change in distance into the slope). If one looks at the slopes of the sides of the pyramids one finds a rather strange collection of values. The Bent Pyramid with two separate slopes, one for the upper portion and one for the lower portion (see figure 1.) has sides with slopes of 1.403, corresponding to an angle of elevation of 54.5 degrees, and 0.950, corresponding to an angle of elevation of 43.5 degrees; see figure 2.

Figure 1. Bent Pyramid (photo from the Brithsh Museum)

Figure 2. Side view of the Bent Pyramid showing the angles of elevation of the lower and upper portions.

At first these seem to be rather odd numbers. The 1.403 is very close to 7/5 which would fit easily into a scheme of determine the slope by the ratio of simple integer multiples of some fixed unit of measure of distance, but the 0.950 is not very close to any simple ratio of integers. If instead of concentrating upon the slopes of the sides, one concentrates on the slopes of the edges of the pyramids, things look a little simpler. The edge of a side is shown in figure 3.

Figure 3. a) Oblique view of a pyramid, b) Transparent oblique view of a pyramid showing the angle of elevation of the edge, theta. The slope of the edge is the distance ed divided by the distance cd.

The slopes of the edges of the Bent Pyramid are 0.992 (very close to 1/1) and 0.672 (very close to 2/3). The pyramid at Dahshur has an edge slope of 0.673 (very close 2/3). The pyramid of Unas has an edge slope of 0.399 (very close to 2/5). The pyramid of Neit has an edge slope of 1.253 (very close to 5/4), and the pyramid of Ammenemes has an edge slope of 0.803 (very close to 4/5). (See the entries for the measured side and edge angles and slopes in table I below.)

Consider the process that the Egyptian builders would have to have gone through in order to build a pyramid with reasonably flat and constant sloped faces. The stones were laid down a single layer at a time. Once one layer was completed (or at least the outer edges of that layer were completed) the starting outer edge of the next layer had to be set back from the edge of the previous layer. One possibility would be measure back a fixed distance from the edge of the previous layer to mark the edge of the next layer. This would involve measuring back two points at the ends of each side of the previous layer, eight separate measurements. See figure 4.

Figure 4. Edges of each new layer (b to f, e to i, h to l and k to c) are laid out by first measuring fixed distances from a to c, a to b, d to e, d to f, g to h, g to i, j to k and j to l, eight separate measurements.

To make sure the slope was kept constant during the building process the set back distance for each layer would have to be some multiple of the height of the next layer of stones. For example if one wants the side slope to be 1.403 then the set back distance for the next layer has to be 1/1.403 or 0.713 time the height of the next layer.

Then lines would be stretched between the pairs of points on each side and stones would then be placed against these lines. In order to make sure the pyramid remained square as new layers were added the builders would probably have measured the diagonal distance between opposite corners of the new layer. If the two diagonal distances were not equal then the next layer would not be square and the lines for the new layer would have to be reset.

Figure 5. Diagonal lines a to c and e to g on a layer of the pyramid. Corners of each new layer are marked by measuring in a fixed distance from the corners along the diagonal lines a to b, g to h, c to d, and e to f, four separate measurements. The edges of the next layer are then laid out b to h, h to d, d to f and f to b. The dashed line is the edge of two adjoining faces of this pyramid.

If the Egyptians had tried to simplify the laying out of the sides for each layer then they would have made the set back distance along the diagonals some simple multiple of the block height of the next layer, thus making the edge slopes simple ratios of integers. This seems to be the case. The measured values for some of the pyramids of Egypt are shown in table I. In some cases the edge slopes appear to be quite close to simple ratios of integers. I have used these simple ratios and developed a theoretical slope for the various pyramids as is shown in the table. The discrepancy between the theoretical values and measured values are also shown. Most of these discrepancies are a few tenths of a degree or less, a reasonable error for naked eye sighting.

Table I. Slopes and edge slopes for some Egyptian pyramids. Side slope data taken from The Pyramids of Egypt by I.E.S. Edwards.

The theoretical edge slope is picked to be a simple ratio if possible, one which give a side angle matching the measured side angle for the given pyramid. The data suggest that the construction method involved determine the edge slope in a simple way. Once the edge slope is fixed the side slope and hence shape of the pyramid is completely determined. The resulting discrepancies between the theoretical edge angles and the measured edge angles are shown in the table.

Early pyramids had simple edge slopes while Cheops and immediate followers used the 2/sqrt(5) slope. Later pyramids revert to the simple ratios. Pepi II is an exception.

The King's Chamber in the Cheops pyramid has a length to width ratio of 2 to 1 and a height to width ratio of 1.11 to 1. Note: sqrt(5)/2 = 1.11803. Thus the sqrt(5)/2 ratio already exists in the pyramid and hence the edge slope of the inverse of this ratio is not that unusual. Sqrt(5) is obviously the hypotenuse of a 1 by 2 right triangle.

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All text copyright ©2002 by C. Hartley, Department of Physics, Hartwick College, Oneonta, New York, unless otherwise noted.