How this calculator works
The apex of a right square pyramid lies directly above the centre of the square. A face slant runs to the midpoint of a base edge; a lateral edge runs to a corner and is longer. Both form right triangles with the vertical height. The four congruent faces each have area half the base edge times the face slant. Reverse area modes recover that slant before finding the height. The surface table prevents confusing one triangular face with the whole lateral surface.
Formula
s² = h² + a²/4; e² = h² + a²/2; A_face = as/2; A_lateral = 2as; A_total = a² + 2as; V = a²h/3.
Worked example
With a = 6 m and h = 4 m, s = √(16 + 9) = 5 m. Each face covers 15 m², all four cover 60 m², and the square base adds 36 m² for 96 m² total. Volume is 36 × 4/3 = 48 m³.
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Frequently asked questions
Is slant height the distance to a corner?
No. Slant height reaches the midpoint of an edge. The lateral edge reaches a corner; the calculator reports both.
What area goes in face mode?
Enter one triangular face only. Choose lateral mode for all four triangles, or total surface mode when the square base is included too.
Why can a small surface area be impossible?
For a positive height the face slant must exceed half the base edge. An area implying a smaller slant cannot form a centred pyramid above that base.