A hekat is a measure of volume defined as 1/30th of a cubic cubit.

RMP #'s 35-38 involve solving for fractions of a hekat.

In "Mathematics in the Time of the Pharoahs", Gillings gives problem #38 as follows:

3 1/7 times the container is required to fill the hekat measure. What is the volume of the container? To find the volume of the container, the scribe has to divide one by 3 1/7.

In other words the ratio between one hekat and the volume of the container is pi:1 and the ratio between the container and one hekat is 1:pi. This is a clear expression of 22/7 and 7/22, contrary to the suggestion that there are no textual references to the 3 1/7 ratio that is found in the great pyramid and elsewhere in the art and architecture of ancient Egypt. One of the lines of RMP 38 is "1/22 divided by 1/7", which produces 7/22,or 1/pi. Keeping in mind the ancient Egyptian prohibition against any fractions other than unit fractions, with the sole exception of 2/3, this is the clearest way that the 7/22 ratio could be textually expressed in ancient Egypt. In problem 38, the scribe goes on to calculate the unit fractios required to express 7/22. Namely:

1/6 plus 1/11 plus 1/22 plus 1/66.

This is the volume of the container that is 3 1/7 times less than one hekat.

By our way of reckoning, the lowest common demoninator is 66:

1/6 = 11/66. 1/11 = 6/66. 1/22 = 3/66. 11/66 plus 6/66 plus 3/66 plus 1/66 equals 21/66 = 7/22.

The ratio between the volume of a cube and the volume of a cylinder with a radius and a height equal to the sidelengths of the cube is 1:pi. In the case of a cylinder equal to one hekat, the radius of the cylinder is 6.152 fingers and the height is also 6.152 fingers. The volume of a cube with sidelengths of 6.152 fingers is one hekat divided by pi.

Here is a webpage that discusses RMP #38.

[www.seshat.ch]

The author mentions that one of the subdivisions of the finger on the cubit rod of Amenhotep I (presently on display in the Louvre) is 1/15th of a finger (This is the 14th finger from the right edge, or the 15th finger from the left edge of the measuring rod). With this division of the finger, 420 units equal one cubit (28 x 15 = 420). A container with dimensions in these units of 84 x 140 x 210 produces a volume of one hekat. These dimensions also produce a cubic diagonal of precisely 266 units, as these dimensions are in the same ratio as a 6-10-15 quadruple, producing a cubic diagonal of 19.

As far as I can tell, there is no cube, or cylinder with a radius equal to it's height, that is in a whole number of units of ancient Egyptian measure, that is equal to one hekat. This makes me wonder about the preferred dimensions of a one hekat measure, whether right angled or cylindrical.