By Joe Foley · Updated 22 September 2026
How palms describe a slope
A seked expresses horizontal run in palms for one royal cubit of vertical rise. With seven palms to the cubit, a seked of 5½ means that the face moves inward 5½ palms while rising seven palms. A larger seked describes a flatter face; a smaller one describes a steeper face. The number measures a proportion, so a small model and a large pyramid can share it.
Take a vertical section through the apex and the midpoint of a square pyramid’s base side. The horizontal leg runs from the centre to the edge, so it is half the base, not the whole base. If that run is r and the height is h, the calculation is seked = 7 × r ÷ h. Both lengths must use the same unit. The result is palms of run per cubit of rise.
The surviving mathematical text gives this definition a concrete setting. In Chace’s translation of Rhind Problem 56, a pyramid has a height of 250 cubits and a base side of 360 cubits. The procedure halves the base, divides that run by the height, and converts the result into palms.1
“A cubit is 7 palms.”
In modern notation, the worked answer is 7 × 180 ÷ 250 = 5.04 palms, or 5 + 1/25 palms. The decimal is our way of displaying the fraction. This example establishes the calculation without requiring metres, degrees or a proposed connection to a famous monument.
Problems 57 and 58 reverse the question: a base and a seked determine a height, then the height and base recover the seked. Chace gives 140 cubits, 5 palms 1 finger, and 93⅓ cubits for that paired exercise. His treatment of Problem 60 is more qualified: its object is translated as a possible pillar and its wording needs interpretation. It should not be grouped casually with the clear pyramid examples.1
One triangle determines all the ratios
For a square pyramid with straight faces, set s = seked. The height-to-half-base ratio is then h/r = 7/s. Its face angle follows from that same ratio. The slant height, measured up the middle of a face rather than along a corner edge, is √(h² + r²).
The base perimeter is eight times the half-base. Dividing by height gives perimeter/height = 8s/7. Dividing slant height by half-base gives slant/half-base = √(1 + (7/s)²). These are dependent results. Change the seked and the perimeter ratio rises while the slant ratio falls. They cannot be tuned independently within this model.
The Great Pyramid preset makes that relationship easy to reproduce. It uses the declared 440-by-280 royal-cubit model, whose half-base is 220 cubits. Petrie discussed the 440-cubit side and 280-cubit height as a theory of the pyramid’s dimensions, separately from his survey.2 The model gives 7 × 220/280 = 5½ palms and a face angle of about 51.84 degrees.
Its perimeter ratio is 1760/280 = 44/7 ≈ 6.285714. The comparison value 2π ≈ 6.283185 is slightly lower: the signed difference, model minus comparison, is about +0.002529, or +0.0402%. The slant ratio is √317/11 ≈ 1.618590, about +0.000556 above the golden ratio φ ≈ 1.618034, or +0.0344%.
Both near-matches come from the same triangle. Counting them as two independent confirmations would count the slope twice. Petrie himself advanced a pi interpretation; that is a named historical argument, not a second measurement. The arithmetic establishes proximity. A claim about what builders intended needs evidence beyond the fit.2
A design model differs from a measured monument
Petrie reported a mean casing base of 9068.8 ± 0.5 inches and a reconstructed height of 5776.0 ± 7.0 inches. These figures describe his reconstruction from surviving evidence; the much larger height uncertainty matters when testing a ratio.2 An exact integer model does not make those uncertainties disappear.
The selector calculates Khufu’s geometry from 440 and 280 cubits. Its rounded metre summary, 230.4 m by 146.6 m, is a separate cross-check, so dividing those displayed metre values will not reproduce every final digit. For the other presets, the tool converts the selected metre dimensions into the site’s working Giza royal cubit. Those are comparative conversions, not claims that each monument was laid out with that exact local standard.
Changing a common length unit leaves the dimensionless ratios unchanged. It changes the numerical lengths, areas and volumes. The unit therefore matters for the size of a result while the selected base-to-height proportion governs the ratio comparisons. Neither conversion nor extra decimal places improves the underlying historical measurement.
What each preset represents
The eight presets compare ideal envelopes of selected construction phases. Their sources do not supply one common set of statistical error bars. “Approximate”, a reported range, and an exact model value describe different kinds of evidence; none should be silently substituted for another.
- Khufu: the original smooth square-pyramid design model, 440 cubits across and 280 high. The integers are exact within the model; Petrie’s reconstructed fabric has the separate uncertainties given above.2
- Khafre: Jones’s 215 m base and 143.5 m height define a smooth square envelope. His construction model is a synthesis, not a claim that the surviving masonry forms a perfect solid. Formal dimensional error bounds are not stated for these inputs.3
- Menkaure: Jones reports a base range of 102.2–104.6 m and a height of 65 m. The tool chooses the 103.4 m average for a square envelope. The reported range describes differing base dimensions; it is not a confidence interval around an ideal square.3
- Djoser: the final P2 phase uses 121 × 109.2 m and approximately 60 m height, following Dickinson’s account of Lauer. Its rectangular footprint gives two face sections. The smooth comparison envelope is not the stepped masonry volume.4
- Bent Pyramid: Jones’s two-slope reconstruction uses a base of 188.60 m, total height 105.07 m, bend height 49.07 m and lower angle 54°31′13″. The upper rise is approximate; formal error bounds are not stated. The tool derives the bend width and calculates each face segment separately.3
- Red Pyramid: a smooth square envelope uses a 220 m base and an original height of about 105 m, as summarised by Jones. These approximate inputs do not establish a precise ancient cubit count.3
- Meidum: UCL’s 144 m base and 92 m height describe the reconstructed final smooth form. They are rounded dimensions, with no formal tolerance on that summary page. They are distinct from the ideal 275-by-175-cubit model discussed in the monument article.5
- Unas: Lehner’s summary table supplies a 57.75 m base and 43 m original height for the smooth square envelope. No formal error bounds accompany those entries. The resulting ratios describe this selected reconstruction.6
Read a comparison label as a distance, not an identity
The calculator compares perimeter/height with 2π. For straight square pyramids it compares slant/half-base with φ, √2 and 5/3, displaying the nearest candidate when its absolute difference is at most 0.5. That broad boundary is a chosen exploratory screen in ratio units. It is not a survey tolerance, confidence interval or probability of ancient intention.
The displayed delta is the absolute gap, so it does not tell you which value is larger. The signed Khufu differences above supply that direction. Also notice how the same 0.5 boundary covers a different percentage of each target: a “match” label cannot rank historical plausibility across the constants.
Try Khufu, then the Red Pyramid. The Red Pyramid’s declared dimensions give a seked of about 7.33 palms and a perimeter/height ratio of about 8.38, far from 2π. Its slant ratio is about 1.38, closer to √2 than to φ. A flatter slope produces a different pair of ratios by the same equations; there is no need to assign every pyramid the same numerical intention.
Then select Djoser and the Bent Pyramid. Djoser needs two rectangular face sections. The Bent Pyramid’s displayed slant follows two segments, so it is not the straight slant used in the square-pyramid formula. The tool withholds named slant comparisons for both. Shape is part of the calculation, not a detail to discard when a ratio looks attractive.
Use the share link to retain a selected monument, then follow its article for the architectural setting. The seked offers a compact way to describe slope; the linked ratios show what that choice entails. Reading the model and the monument together keeps the mathematical relationship clear without turning it into an unwritten building instruction.
Sources
- Arnold Buffum Chace, The Rhind Mathematical Papyrus, vol. I (1927), pp. 96–99, problems 56–60. Free translation and commentary; the worked example and quotation use pp. 96–97. Source register.
- W. M. Flinders Petrie, The Pyramids and Temples of Gizeh (1883), p. 43 for the reconstructed envelope and p. 183 for the 440-by-280 theory and his pi interpretation. Digital Giza record.
- Richard Jones, The Use of Manpower in the Construction of Old and Middle Kingdom Pyramid Complexes (2006), pp. 96, 109, 170–171 and 194. Construction-model synthesis and its attributed dimensions. University of Liverpool record.
- Tessa Dickinson, A Landscape and Materials-based Approach to Royal Mortuary Architecture in Early Third Millennium BC Egypt, vol. I (2014), p. 146, following Lauer. UCL record.
- UCL Digital Egypt, Meydum pyramid: further information, dimensions and construction phases. Read the record.
- Mark Lehner, The Complete Pyramids (1997; 2001 reprint), p. 17, “Major Pyramid Statistics”. Unas original-envelope dimensions. Source register.