How this calculator works
The apex must lie directly above the square’s centre. A face slant height reaches a base-side midpoint, giving a right triangle with horizontal leg a/2. A lateral edge reaches a corner, whose horizontal distance from the centre is a/√2. The table distinguishes these triangles. Base-area inputs first recover a. Area pairs recover the slant from the four matching triangular faces. Volume gives the perpendicular height directly. Only positive, non-flat pyramids are accepted.
Formula
B=a²; s²=H²+a²/4; d²=H²+a²/2; one face F=as/2; L=4F=2as; S=B+L; V=BH/3. From s,d: a=2sqrt(d²-s²). Half the apex angle within one triangular face is atan(a/(2s)).
Worked example
For a 6 m base edge and 5 m slant, H=√(25-9)=4 m. Each face has area 15 m², lateral area is 60 m², total area is 96 m² and volume is 48 m³. The lateral edge is √34 m, which is longer than the face slant.
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Frequently asked questions
Which sloping length did I measure?
A line from the apex to a side midpoint is the face slant height. A line from the apex to a base corner is a lateral edge. Choose the corresponding pair.
Can I recover the height without knowing the base?
Yes. Slant plus lateral edge, slant plus face area, and lateral plus total area are examples of pairs that also recover the base edge.
Does this work for an off-centre apex?
No. The right-triangle and equal-face relationships require a right square pyramid. An oblique pyramid needs additional geometry.