A northern pyramid at Dahshur
Sneferu's Red Pyramid stands north of the Bent Pyramid at Dahshur. Its modern name comes from the reddish limestone core exposed beneath the lost outer surface. The broad, shallow outline invites a simple description in cubits. The difficulty is deciding whether a tidy pair of counts describes its approximate dimensions or independently recovers an ancient design.1
Richard Jones adopts an original height of about 105 metres and a base side of 220 metres for his construction model. These describe a reconstructed outer shape, or envelope. They are the inputs used here and in the Red Pyramid calculator; they are not a new survey or a complete account of the surviving masonry.1 First establish the shape they imply, then test what a change of unit can do.
One shallow face from two approximate lengths
A vertical section through the apex and the midpoint of a base side has a horizontal run of 110 metres, half the base, and a rise of 105 metres. The rise/run ratio is therefore 105 / 110, or 21/22. Taking its arctangent gives a face angle of about 43.67 degrees. This is calculated precision: the approximate lengths do not establish the face to a hundredth of a degree.
Red Pyramid
A seked expresses the horizontal run in palms for one cubit of rise. With seven palms to a cubit, the same triangle gives 7 × 110 / 105=7⅓ palms. Chace's translation of Rhind Problem 56 supplies this half-base-and-rise method and the seven-palm relation.2 Applying that later mathematical notation describes this reconstruction; it does not identify the instructions used by Sneferu's builders.
Two dimensions test a proposed cubit
For comparison, the site's selected Giza royal cubit is 0.523748 metre. Dividing the two inputs by that working factor gives about 420.0 cubits across and 200.5 cubits high.3 Those are modern conversions. To test a simpler possibility, suppose the intended counts were exactly 420 across and 200 high. This is a declared comparison, not an ancient specification.
Fit the base first: 220 metres divided by 420 gives a cubit of about 523.81 millimetres. Fit the height independently: 105 divided by 200 gives 525.00 millimetres. The two implied cubits differ by about 1.19 millimetres.
| Dimension | Metre input | Test count | Implied cubit |
|---|---|---|---|
| Base side Approximate reconstruction | 220 m | 420 | 523.81 mm |
| Original height Approximate reconstruction | 105 m | 200 | 525.00 mm |
Population: two dimensions of one reconstructed envelope. Inputs: Jones, p. 109. Counts: the hypothetical 420:200 comparison. Implied cubits: our divisions, rounded for checking; the extra digits are not measurement accuracy. Jones gives no formal tolerance for the input pair.
Changing the ruler cannot change the proportion
The apparent disagreement is about shape. Divide height by base and the common unit cancels. The metre inputs give 105/220=21/44; the hypothetical counts give 200/420=10/21. The first ratio is about 0.23% larger than the second, whatever length is assigned to a common cubit.
Keep the base at 220 metres and the count at 420 cubits. A height of 200 of those same cubits would be 220 × 200 / 420, or about 104.76 metres: roughly 0.24 metre below the 105-metre input. A different ruler scales both dimensions together and cannot remove that proportional difference.
This distinguishes the two declared models exactly as arithmetic. It cannot decide between historical designs at that small difference: the reported dimensions are approximate and carry no formal survey-error range. The 420:200 pair is neither confirmed nor ruled out as an ancient design by this test. Recovering an ancient ruler would require evidence beyond choosing counts to fit this envelope.
A precise course average can repeat an assumption
The same care applies to the masonry. Jones uses 160 courses and divides the assumed 105-metre height by that count, obtaining an average course height of 0.65625 metre. His discussion distinguishes the mathematical model from course heights that vary through the building.1
Multiplying 160 by 0.65625 returns 105 metres because that height produced the average in the first place. It is not a second measurement confirming the height, nor evidence that every course followed one module. The face angle, cubit counts and model course average all inherit their limits from the inputs.
What the Red Pyramid's measures establish
The reconstructed envelope describes a broad base and a shallow face. It lets us calculate a slope and test a proposed proportion, while keeping the length of a ruler separate from the shape it measures. A tidy count and a long decimal answer each need an independent evidential basis before becoming a claim about the building.
Explore the Red Pyramid calculator preset with the same 220-by-105-metre inputs. Then compare the Bent Pyramid's two separate slopes or Meidum's reconstructed envelope and ideal comparison. The royal cubit dossier introduces the selected comparative standard.
Sources and Further Reading
- Richard Jones, The Use of Manpower in the Construction of Old and Middle Kingdom Pyramid Complexes (University of Liverpool, 2006), pp. 109–110. Site identity, approximate envelope and chosen course model; a secondary construction analysis. Repository record.
- Arnold Buffum Chace, The Rhind Mathematical Papyrus, vol. I (1927), p. 96, Problem 56. Later mathematical method and seven palms per cubit; not direct testimony about the Red Pyramid. Source register.
- Ancient Measures, length conversion matrix, selected Giza royal cubit, 0.523748 m. Used only as a declared comparative factor; it supplies neither a Red Pyramid ruler nor a survey tolerance. Converter source register.