Everybody would have understood by now that a pyramid is a matter of angles.

However, the angles of the pyramid have ordinary values compared to what we are used to. For example, the angle of the inner slopes of the pyramid of Cheops is very close to 26°56’, and the faces’ angle is close to 51°84’ with the horizontal, and the angle of the edges is close to 52° with the horizontal in the median plane.

This is due to the way ancient Egyptians measured angles indirectly using the cotangent of the angles, the , or seked; they did not use degree. This information comes from the Middle Kingdom of Egypt, but the sine, the cosine or else the tangent could have been used from time to time.

More precisely, they used to express length in Palm and height in Cubits, so an SDK was 7 * cotangent of the same angle.

So the chosen angles had at least a value, whether it was the sine, the cosine or the tangent, as simply as possible using the measure scale based on the royal cubit for which the smallest subdivision is the finger, i.e. 1/28.

As for numerical writing, the Ancient Egyptians used fractional notation and not decimal notation.

We can then expect that one, or several, of these trigonometric values of an Egyptian angle will be expressed as a whole multiple of the fraction 1/28.

Many archaeologists have endeavored to record the pyramids’ angles in degrees, minutes and seconds. They better should have recorded the cotangent, the sine or cosine of these angles, which is a ration expressed by the Egyptian fraction for length based on the royal cubit; for example, a 45° angle would equal a seked of 7.

If they had proceeded that way, as they did for the lengths expressed by royal cubits, measurements would have been more meaningful.

The royal cubit (MH NSWT)

- The royal cubit is divided into 28 segments of 1 digit.
- The palm is 4/28 or 1/7; the double palm is the double.
- The small span is 12/28 or 3/7, which equals 3 palms.
- The djeser is 16/28 or 4/7, which equals 4 palms.
- The remen is 20/28 or 5/7, which equals 5 palms.
- The short cubit is 27/28 or 6/7, which equals 6 palms.

As a result, when an angle is found in a pyramid, its SKD should be checked to ensure that it can be expressed in fractions of the units of measurement.

Let’s take, for example, the angle of the face with the horizontal measured by Petrie which used different measurement methods that all gave a different result:

In conclusion, he suggested retaining the value 51°52’ with an error of more or less 2’, i.e., an interval between 50’ and 54’.

He deduced the height from this value, after measuring the base.

Petrie measurements for the base give 9068.8 inches, or 230.34 meters, which, with the addition of a royal cubit of 0.5235 m (1,717 ft.), would give 440 cubits exactly, so the height is 220 * tangent 51°52’, i.e., 280 or 24 cubits.

It seems the constructors made sure the base measurement was a whole number, so we could venture that they did the same for the height, i.e., 280 cubits. Consequently, the tangent or seked would be 22 * 7/28, or 11/2, which probably could be expressed by the value 5 + a half-cubit or big span.

To conclude, in our notation, the angle is 51°50’35’’ as the nearest approximation, which falls within the interval given by Petrie.

However, this value is theoretical because the faces’ median is “hollowed” by 2 cubits from the base, giving the real SKD value at the median level to be 7 * (220-2)/280, or 109/20, or 5 + 9/20, or else 5+1/4+1/5 to use the old Egyptian notation system, or 52°5’48’’ in our current notation system.

*We can guess that any angle of the pyramid is at least a trigonometric function (7 x cotangent) expressed by a whole multiple of the fraction 1/28, in other words, a whole number of “digits”.*

The angle of the slopes is 26°56’ with the horizontal, i.e., a SKD of 14.

Throughout the construction time, there were two days per year when the shadows on the North, East and West faces, as well as the shadows on the north edges, were invisible, as if they “swallowed their shadows”. First, it permitted to position the pyramids in South North and East West with great accuracy using a model, and also to check the correct pyramid alignment thanks to the faces and edges’ shadows, so the edges would meet at the summit, invisible until the last day.

The sun was thus used as a giant instrument of measure to the pyramid’s scale!

Furthermore, the 51.8° angle gives a ratio of 22/7 between the base perimeter and the height, a value near π to 1 per thousand, and gives a ratio between the apothem of one face and the half-base close to the golden ratio with higher accuracy, which is no coincidence!