How this calculator works
The apex must sit directly above the centre of the square. A cross-section through the apex and the midpoint of a base edge makes a right triangle with legs h and a/2. Its hypotenuse is the face slant height, not the corner edge. Each triangular face has area a times that slant height divided by two. Reverse mode subtracts the floor area and divides the remainder between the four faces before recovering vertical height.
Formula
l² = h² + (a/2)²; face = al/2; lateral = 2al; total S = a² + 2al; l = (S-a²)/(2a); V = a²h/3.
Worked example
With base edge 6 m and vertical height 4 m, the slant height is 5 m. Each face covers 15 m². The four faces contribute 60 m² and the base contributes 36 m², for 96 m² altogether.
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Frequently asked questions
Does total area include the floor?
Yes. For an open-bottom model use the lateral area, which counts only the four triangular faces.
Can the apex be off-centre?
These equations assume a centred apex. An oblique pyramid generally has four different face heights.
Why must total area exceed twice the base area?
A positive vertical height makes the slant height greater than a/2. Substituting that bound into a² + 2al gives S > 2a².