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Square Pyramid Height from Seventeen Measurement Pairs

Square Pyramid Height from Seventeen Measurement Pairs. Use unit controls and documented positive geometric roots to compare base, lateral and total measures.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Example in initial displayed units: Base edge (a) 6, Slant height (s) 5 gives Pyramid height (H) 4, Base diagonal 8.4852814, Base area (A_b) 36, Lateral edge (d) 5.8309519, Lateral face area (A_f) 15, Total lateral area (A_l) 60, Total surface area (A) 96, Volume (V) 48.

H² = s²−(a/2)²; H² = d²−a²/2; H = 3V/a²

Worked example

Enter these known values and leave the other values blank.

Find the pyramid height (H) using...
base edge (a) & slant height (s).
Base edge (a)
6 cm
Slant height (s)
5 cm
Pyramid height (H)
4 cm

Assumptions and limitations

  • The apex sits above the center of a square base. Perpendicular height H reaches the base center; slant height s reaches a base-edge midpoint; lateral edge d reaches a corner. Base area is one square and lateral area is the sum of four triangular faces. Select a measurement pair to show its controls. Custom mode exposes additional dimensions; redundant measurements do not determine a unique height.
  • Select compatible units and enough independent dimensions. Clear unknown values before solving backward. Numeric precision and display rounding do not quantify uncertainty in measured inputs.
  • Arithmetic uses 64 significant digits. Original reference diagrams explain the dimensions and do not follow entered proportions.

Common questions

Can I substitute lateral edge for slant height?

No. Lateral edge reaches a corner and exceeds slant height, which reaches the midpoint of a base edge. Select the matching measurement pair.

References

Require physical dimensions and real positive inverse geometry; distinguish dependent measurements and cross-section consistency. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor. The US fluid ounce uses one 128th of the exact US liquid gallon, replacing the rounded captured reciprocal factor. Bookify replaces the captured rounded US gallon with the exact 3.785411784-liter definition.

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