By Joe Foley · Updated 12 October 2026
One slope creates all three results. In the familiar 440 by 280 royal-cubit model of Khufu's pyramid, the half-base is 220 cubits. That gives a 5½-palm seked, a perimeter-to-height ratio near 2π, and a slant-to-half-base ratio near the golden ratio φ. The arithmetic is exact within the declared model; the two famous near-matches are not independent evidence of two further design choices.
01 · BasisBegin with a model, then check the monument
Flinders Petrie reconstructed a mean original casing side of 9,068.8 inches from surviving casing and socket evidence. He gave about 0.5 inch as the probable error of that mean. His original-height estimate of 5,776 inches came from the casing angle and carried a wider uncertainty of about 7 inches. These are survey and reconstruction results, not ancient cubit counts.1
Petrie also discussed a separate theory of a 440-cubit side and 280-cubit height. Those round integers make a useful worked model, but their neatness must not be transferred back to the irregular surviving structure.1 For a fuller account of the survey and its uncertainty, see the Great Pyramid study. Here the integers are inputs to a question about mathematical dependence.
02 · MethodCalculate the 5½-palm seked
Take a vertical section through the apex and the midpoint of one base side. Its horizontal run is half the base, 440 ÷ 2 = 220 cubits; its rise is 280 cubits. An Egyptian seked expresses horizontal run in palms for a rise of one cubit. With seven palms to a cubit, the model gives 7 × 220 ÷ 280 = 5½ palms. A larger seked means a flatter face.
That method has a textual control outside Khufu's monument. In Rhind Problem 56, the scribe starts with a 360-cubit base and 250-cubit height, halves the base and converts the run-to-rise result into palms. The calculation is the same kind of operation; the problem does not identify Khufu's pyramid.2
03 · CalculationDerive both near-matches from the same triangle
The square base has perimeter 4 × 440 = 1,760 cubits. Divide by the height: 1,760 ÷ 280 = 44/7 ≈ 6.285714, about 0.040% above 2π. The face's slant height follows from the right triangle: √(280² + 220²) cubits. Divide that by its 220-cubit horizontal leg to get √317/11 ≈ 1.618590, about 0.034% above φ. These are modern comparisons of a model, not measurements of inscribed constants.
| Seked | Perimeter ÷ height | Slant ÷ half-base | Basis |
|---|---|---|---|
| 5½ palmsKhufu model | 6.285714 near 2π | 1.618590 near φ | 440-cubit side, 280-cubit height |
| 6 palmsIllustrative change | 6.857143 | 1.536591 | Same square-pyramid formula; no monument attribution |
The second row is a mathematical control, not a proposed ancient design. It changes only the slope: for any seked s in palms, perimeter/height = 8s/7 and slant/half-base = √(1 + (7/s)²). One result rises as the other falls. Once s is fixed, neither can be selected independently.
04 · InterpretationA close fit needs a separate historical argument
Petrie advanced a π reading when discussing the 440-cubit theory. That is evidence of Petrie's interpretation, not proof of a Fourth Dynasty instruction to encode π. The φ comparison is another modern observation produced by the same triangle. Neither near-match by itself identifies a builder's purpose.1
The worked example shows what a ratio match can establish: a relationship inside a declared model. To change the inputs and compare other pyramid reconstructions, use the pyramid calculator. To test how a visually persuasive match can overstate its evidence, continue with False Precision.
05 · SourcesSources
- W. M. Flinders Petrie, The Pyramids and Temples of Gizeh (London: Field & Tuer; New York: Scribner & Welford, 1883), pp. 78–82 (printed pp. 39–43) for the casing-base and height reconstruction; p. 222 (printed p. 183) for the 440-by-280-cubit theory and Petrie's π reading. Digital Giza catalogue record.
- Arnold Buffum Chace et al., The Rhind Mathematical Papyrus, vol. I (Oberlin: Mathematical Association of America, 1927), pp. 112–113 (printed pp. 96–97), Problem 56.